id int64 | problem_number string | title string | statement string | background string | difficulty_level_id int64 | status string | proposed_by string | proposed_year int64 | category_id int64 | set_id int64 | view_count int64 | favorite_count int64 | created_at timestamp[s] | updated_at timestamp[s] | published bool | category dict | difficulty dict | set dict | duplicates list | status_review unknown | archive_reason string | duplicate_of list | old_problem_number string | notes string | tags list | literature_assessment string | literature_sources list | literature_checked_at timestamp[s] | solution_reference string | source_url string | source string | prize string | references list | clean_statement string | original_statement string | statement_status string | statement_verification string | research_classification string | research_summary string | research_difficulty_suggested string | historical_research_summary string | research_summary_reviewed_at timestamp[s] | source_citation string |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2,564 | KOU-21.55 | Kourovka Notebook Problem 21.55 | Let $q$ be a power of a prime $p$, and let $m_n(q)$ be the maximum $p$-length of $p$-solvable subgroups of $\operatorname{GL}(n,q)$. Is it true that $\lim_{n\to\infty}m_n(q)/\log_2 n=1$? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.55.
Discussion and literature:
I. G\"ulo\u{g}lu
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** No proof was found t... | 2 | open | \.{I | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,565 | KOU-21.56 | Kourovka Notebook Problem 21.56 | Let $\ell(X)$ denote the composition length of a finite group $X$. Let $A$ be a finite nilpotent group acting by automorphisms on a finite soluble group $G$. Let $c(G,A)$ be the number of trivial $A$-modules in a given $A$-composition series of $G$. (Note that $c(G,A)=\ell(C_G(A))$ if $(|A|,|G|)=1$.)
Conjecture: there... | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.56.
Discussion and literature:
I. G\"ulo\u{g}lu
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_solved
**Classification:** PARTIAL-PROGRESS
**Current literature assessment.** The... | 3 | partially_solved | \.I. G\"ulo\v{g | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,566 | KOU-21.57 | Kourovka Notebook Problem 21.57 | Let X be a non-empty class of finite groups of odd order closed under taking subgroups, homomorphic images, and extensions. Let H be an X-maximal subgroup of a finite group G, and N a normal subgroup of G. Must H $\cap$ N be an X-maximal subgroup of N? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.57.
Discussion and literature:
W. Guo, D. O. Revin
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_solved
**Classification:** PARTIAL-PROGRESS
**Current literature assessment.** ... | 2 | partially_solved | W. Guo, D. O. Revin | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,567 | KOU-21.58 | Kourovka Notebook Problem 21.58 | We say that a product $XY=\{xy\mid x\in X,\ y\in Y\}$ of two subsets $X,Y$ of a group $G$ is direct if for every $z\in XY$ there are unique $x\in X$, $y\in Y$ such that $z=xy$. Is there an infinite group $G$ such that every subset $A\subseteq G$ satisfies the following property: all the maximal subsets $B$ for which th... | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.58.
Discussion and literature:
Note that for checking the property for a given infinite group $G$, it suffices to consider only those subsets $A\subseteq G$ for which $|A|=|G\setminus A|$. Indeed, the property is equivalent to $A^{-... | 2 | solved | M. H. Hooshmand | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] No, there are no such groups (M. I. Kabenyuk, *Preprint*, February 2026, https://arxiv.org/abs/2602.2287... | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,568 | KOU-21.59 | Kourovka Notebook Problem 21.59 | For a finite group $G$, let $\chi_1(G)$ denote the totality of the degrees of all irreducible complex characters of $G$ with allowance for their multiplicities. Suppose that $H$ is a finite group with $\chi_1(H)=\chi_1(G)$.
a) If $G$ is an almost simple group, must $H$ be isomorphic to $G$?
b) If $G$ is a quasisimple... | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.59.
Discussion and literature:
This is true if $G$ is a simple group (see 11.8(a) in Archive). A. Iranmanesh, F. Shirjian
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_solved
**... | 2 | partially_solved | A. Iranmanesh, F. Shirjian | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "partially_solved",
"checked_at": "2026-10-16",
"notes": "Subproblem confirmed solved (*) in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] b) Yes, it must, by Theorem B in (C. Bessenrodt, H. N. Nguyen, J. B. Olsson, H. P. Tong-Viet, ... | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,569 | KOU-21.60 | Kourovka Notebook Problem 21.60 | Let $G$ be a finite group, $\mathbb Z_{(p)}$ the localization at $p$, and $\mathbb F_p$ the field of $p$ elements. Let $\mathcal X$ be the class of $\mathbb F_pG$-modules obtained by reduction of simple $\mathbb QG$-modules. Is it true that $\mathbb Z_{(p)}G$ is semiperfect if and only if each projective indecomposable... | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.60.
Discussion and literature:
The "only if" direction is proved, and the affirmative answer is obtained if $p$ does not divide $|G|$ (D. Johnston, D. Rumynin, J. Algebra, 687, no. 1 (2026), 776--791). D. Johnston, D. Rumynin
<!-- ... | 2 | solved | D. Johnston, D. Rumynin | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] No, not always (AI solution confirmed by D. Rumynin: \\ M. Kinyon, J. Urban, *Preprint*, 2026,\nhttps://... | null | null | null | [Discussion #4 literature review, 2026-10-14] **solved (no)**; the dataset currently labels it partially solved. D. Johnston, D. Rumynin: is β€βββG semiperfect if and only if each projective indecomposable π½βG-module is an β-combination of reductions of simple βG-modules in the Grothendieck group (the "only if" directi... | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,570 | KOU-21.61 | Kourovka Notebook Problem 21.61 | For a fixed (finitely generated free)-by-cyclic group $G=F_n\rtimes\mathbb Z$, is there an algorithm that, given a finite subset $S$ of $G$, finds a finite presentation for the subgroup $H=\langle S\rangle$? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.61.
Discussion and literature:
Cf. 4.8 in Archive. I. Kapovich
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** No un... | 2 | open | I. Kapovich | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,571 | KOU-21.62 | Kourovka Notebook Problem 21.62 | Is the uniform subgroup membership problem decidable for (finitely generated free)-by-cyclic groups? That is, for a fixed group $G=F_n\rtimes\mathbb Z$, is there an algorithm that, given elements $w,h_1,\ldots,h_k\in G$, decides whether or not $w$ belongs to the subgroup $H=\langle h_1,\ldots,h_k\rangle$? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.62.
Discussion and literature:
I. Kapovich
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** No uniform subgroup-membe... | 2 | open | I. Kapovich | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,572 | KOU-21.63 | Kourovka Notebook Problem 21.63 | Let $F$ be a field of characteristic $p>0$, and let $\Gamma$ be the principal congruence subgroup of $\operatorname{Aut}(F[x_1,\ldots,x_n])$ consisting of all automorphisms that send each variable $x_i$ to $x_i$ modulo terms of higher degree. Then $\Gamma$ is a residually-$p$ group. Does $\Gamma$ satisfy a pro-$p$ iden... | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.63.
Discussion and literature:
(E. Zelmanov).
E. I. Khukhro
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_solved
**Classification:** PARTIAL-PROGRESS
**Current literature asse... | 2 | partially_solved | E. I. Khukhro | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,573 | KOU-21.64 | Kourovka Notebook Problem 21.64 | Is it true that if a normal subgroup $A$ of a Sylow $p$-subgroup of a $p$-soluble finite group $G$ has exponent $p^e$, then the normal closure of $A$ in $G$ has $(p,e)$-bounded (or even $e$-bounded) $p$-length? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.64.
Discussion and literature:
E. I. Khukhro
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** No p-length bound from ... | 2 | open | E. I. Khukhro | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,574 | KOU-21.65 | Kourovka Notebook Problem 21.65 | Suppose that $\phi$ is an automorphism of a finite soluble group $G$. Must $G$ contain a subgroup of index bounded in terms of $|\phi|$ and $|C_G(\phi)|$ whose Fitting height is bounded
(a) in terms of $|\phi|$?
(b) or even in terms of the composition length of $\langle\phi\rangle$? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.65.
Discussion and literature:
An affirmative answer to part (b) is known when $|\phi|$ is a prime power (B. Hartley--V. Turau), or when $(|G|,|\phi|)=1$ (A. Turull, B. Hartley--I. M. Isaacs). Also cf. 19.43. E. I. Khukhro
<!-- LIT... | 2 | partially_solved | E. I. Khukhro | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,575 | KOU-21.66 | Kourovka Notebook Problem 21.66 | Suppose that A is a nilpotent group of automorphisms of a finite soluble group G. Is the Fitting height of G bounded in terms of |A| and |CG(A)|?
Isaacs), when A is cyclic (see 19.43), or when CG(A) = 1 (E. C. Dade). | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.66.
Discussion and literature:
An affirmative answer is known when (|G|, |A|) = 1 (J. G. Thompson, even for soluble A, with improved bounds in subsequent papers of H. Kurzweil, A. Turull, B. Hartley--I. M.
Note that for any non-nil... | 2 | partially_solved | E. I. Khukhro | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,576 | KOU-21.67 | Kourovka Notebook Problem 21.67 | Suppose that $\phi$ is an automorphism of a finite soluble group $G$, and let $r$ be the (Pr\"ufer) rank of the fixed-point subgroup $C_G(\phi)$. Is the Fitting height of $G$ bounded in terms of $|\phi|$ and $r$? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.67.
Discussion and literature:
This is known to be true in the case where $|\phi|$ is a product of at most two prime powers (B. Hartley, Preprint, 1994, https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2025/08/hartley94-... | 2 | partially_solved | E. I. Khukhro | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,577 | KOU-21.68 | Kourovka Notebook Problem 21.68 | A finite group $G$ is said to be semi-abelian if it has a sequence of subgroups $1=G_0\leqslant G_1\leqslant\cdots\leqslant G_n=G$ such that for every $i$ the subgroup $G_{i+1}$ is isomorphic to a quotient of a semidirect product $A_i\rtimes G_i$ for some abelian group $A_i$.
Conjecture: Semi-abelian finite groups are... | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.68.
Discussion and literature:
M. Kida
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-09-27)
**Status:** solved
**Classification:** SOLVED-IN-LITERATURE
**Current literature assessment.** Alper Ferudun report... | 3 | solved | M. Kida | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] Conjecture is refuted in (A. Rizzoli, *Preprint*, 2026,\n\nhttps://kourovkanotebookorg.wordpress.com/wp-... | null | null | null | null | null | Alper Ferudun reports a semi-abelian group of order 768 with a non-monomial irreducible character of degree 8, answering Kidaβs question negatively. This is an unrefereed, AI-assisted preprint, with author-reported exact computations and an AI audit; no minimal-order claim or independent human verification is made.
**... | [
{
"url": "https://doi.org/10.5281/zenodo.23000305",
"label": "Alper Ferudun, A Semi-Abelian Group of Order 768 That Is Not an M-Group (2026)"
},
{
"url": "https://eulersolve.org/papers/kou-21-68/",
"label": "EulerSolve paper and computational materials"
},
{
"url": "https://huggingface.c... | 2026-09-27T00:00:00 | Alper Ferudun, A Semi-Abelian Group of Order 768 That Is Not an M-Group (2026): https://doi.org/10.5281/zenodo.23000305
EulerSolve paper and computational materials: https://eulersolve.org/papers/kou-21-68/ | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,578 | KOU-21.69 | Kourovka Notebook Problem 21.69 | Is there an algorithm deciding if a given one-relator group is hyperbolic? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.69.
Discussion and literature:
D. Kielak
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** No algorithm deciding hyper... | 2 | open | D. Kielak | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,579 | KOU-21.70 | Kourovka Notebook Problem 21.70 | A group $G$ is called an orientable Poincar\'e duality group of dimension $n$ over a ring $R$ if it is of type $FP$ over $R$ and $H^i(G;RG)=0$ for $i\ne n$, while $H^n(G;RG)=R$ as an $RG$-module, where the action on $R$ is trivial. (Note that $G$ is not required to be finitely presented.) If $G$ is an orientable Poinca... | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.70.
Discussion and literature:
D. Kielak
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** No integral Poincare-dualit... | 2 | open | D. Kielak | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,580 | KOU-21.71 | Kourovka Notebook Problem 21.71 | For a ring $R$, we say that a group $G$ is of type $FL(R)$ if the trivial $RG$-module $R$ admits a finite resolution by finitely generated free modules. If $R$ is a field, we define the Euler characteristic of $G$ over $R$ to be the alternating sum of $R$-ranks of homology groups $H_i(G;R)$. Does there exist a group of... | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.71.
Discussion and literature:
D. Kielak
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** No group of the stated two-... | 2 | open | D. Kielak | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,581 | KOU-21.72 | Kourovka Notebook Problem 21.72 | We say that a group is a Tarski monster if it is finitely generated, not cyclic, and all of its proper non-trivial subgroups are isomorphic to each other.
a) Do there exist amenable torsion-free Tarski monsters?
b) Do there exist amenable Tarski monsters of prime exponent? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.72.
Discussion and literature:
D. Kielak
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** No amenable torsion-free or... | 2 | open | D. Kielak | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,582 | KOU-21.73 | Kourovka Notebook Problem 21.73 | Is the conjugacy problem in $\operatorname{CT}(\mathbb Z)$ algorithmically decidable? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.73.
Discussion and literature:
See the definition of $\operatorname{CT}(\mathbb Z)$ in 17.57. S. Kohl
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
*... | 2 | open | S. Kohl | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,583 | KOU-21.74 | Kourovka Notebook Problem 21.74 | Is it algorithmically decidable whether a given element $g\in\operatorname{CT}(\mathbb Z)$
(a) permutes a nontrivial partition of $\mathbb Z$ into residue classes?
(b) has only finite cycles?
(c) has no finite cycles? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.74.
Discussion and literature:
S. Kohl
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** No algorithms deciding the th... | 2 | open | S. Kohl | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,584 | KOU-21.75 | Kourovka Notebook Problem 21.75 | Given two distinct sets $P_1$ and $P_2$ of odd primes none of which is a subset of the other, is it true that $\langle \operatorname{CT}_{P_1}(\mathbb Z),\operatorname{CT}_{P_2}(\mathbb Z)\rangle \lneq \operatorname{CT}_{P_1\cup P_2}(\mathbb Z)$? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.75.
Discussion and literature:
See the definition of $\operatorname{CT}_P(\mathbb Z)$ in 17.60. S. Kohl
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
... | 2 | solved | S. Kohl | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] No, in fact, equality holds (A. L. Iskra, *Preprint*, 2026,\nhttp://arxiv.org/abs/2610.04274). A simplif... | null | null | null | [Discussion #4 literature review, 2026-10-14] **solved (no; equality holds)**; the dataset currently labels it open. S. Kohl: is β¨CT_{Pβ}(β€), CT_{Pβ}(β€)β© a proper subgroup of CT_{PββͺPβ}(β€)? Notebook v48: β21.75, "No, in fact, equality holds", with a reference to A. Iskra, Groups of class transpositions with prescribed ... | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,585 | KOU-21.76 | Kourovka Notebook Problem 21.76 | Let $\sigma=(\sigma_{ij})$, $1\leqslant i\ne j\leqslant n$, be an irreducible elementary net (carpet) of order $n\geqslant 3$ over a field $K$ (see 19.48). The net $\sigma$ is said to be closed if the elementary net subgroup $E(\sigma)$ does not contain new elementary transvections. The net $\sigma$ is said to be compl... | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.76.
Discussion and literature:
Completable elementary nets are closed. It is known that over fields of characteristic 0 and 2 there exist irreducible closed elementary nets that are not completable (V. A. Koibaev, Trudy Inst. Mat. M... | 2 | solved | V. A. Koibaev | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-09-28T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] Yes, such nets do exist (Ya. N. Nuzhin, *Siberian Math. J.*, **67**, no. 4 (2026), 840--845)."
} | null | null | null | null | null | Nuzhinβs Corollary 1 constructs irreducible closed elementary matrix carpets that cannot be completed, in every characteristic and every degree at least 2. This includes the requested odd-characteristic examples of order at least 3. Ferudunβs later unrefereed, AI-assisted note supplies explicit examples and one-variabl... | [
{
"url": "https://doi.org/10.1134/S0037446626040099",
"label": "Ya. N. Nuzhin, Carpet Lie Rings Not Invariant with Respect to Carpet Subgroups, Siberian Math. J. 67 (2026), Corollary 1"
},
{
"url": "https://doi.org/10.5281/zenodo.23004034",
"label": "Alper Ferudun (2026), Explicit Closed Element... | 2026-09-28T00:00:00 | Ya. N. Nuzhin, Carpet Lie Rings Not Invariant with Respect to Carpet Subgroups, Siberian Math. J. 67 (2026), Corollary 1: https://doi.org/10.1134/S0037446626040099
Alper Ferudun (2026), Explicit Closed Elementary Nets That Cannot Be Completed β unrefereed preprint: https://doi.org/10.5281/zenodo.23004034
EulerSolve man... | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,586 | KOU-21.77 | Kourovka Notebook Problem 21.77 | Let $d$ be an integer that is not divisible by $n$-th powers of primes, let $x^n-d$ be an irreducible polynomial over $\mathbb Q$, let $\theta=\sqrt[n]{d}$, and let $K=\mathbb Q(\theta)$ be the radical extension of degree $n$ of the field $\mathbb Q$. The multiplicative group $K^*$ of the field $K$ is canonically embed... | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.77.
Discussion and literature:
Background on rich elementary transvections and the associated net: (V. A. Koibaev, St. Petersbg. Math. J., 21, no. 5 (2010), 731--742); (V. A. Koibaev, A. V. Shilov, J. Math. Sci. New York 171, no. 3 ... | 2 | open | V. A. Koibaev | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,587 | KOU-21.78 | Kourovka Notebook Problem 21.78 | Let $p$ be a prime and let $G$ be a pro-$p$ group. Suppose that all of the (continuous Galois) cohomology groups $H^n(G,\mathbb F_p)$ of $G$ with coefficients in the field of $p$ elements are finite. Does it necessarily follow that the cohomology ring $H^*(G,\mathbb F_p)$ is finitely generated? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.78.
Discussion and literature:
The answer is known to be `yes' if $G$ is abelian-by-($p$-adic analytic), as follows from (J. King, Commun. Algebra, 27, no. 10 (1999), 4969--4991). P. Kropholler
<!-- LITERATURE-TRIAGE:BEGIN -->
## L... | 2 | partially_solved | P. Kropholler | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,588 | KOU-21.79 | Kourovka Notebook Problem 21.79 | Let $G$ be a finitely generated group with a fixed finite generating set $S$ and the corresponding word metric $L_S(*)$. An element $g$ is said to be distorted in $G$ if $L_S(g^n)/n\to 0$ as $n\to\infty$; this notion is independent of the choice of the generating set $S$. For any, not necessarily finitely generated, gr... | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.79.
Discussion and literature:
Note that it is straightforward to construct countable (not finitely generated) left-orderable groups with this property using HNN-extensions and applying results of V. V. Bludov and A. M. W. Glass. Y.... | 2 | open | Y. Lodha, A. Navas | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,589 | KOU-21.80 | Kourovka Notebook Problem 21.80 | Do there exist finitely generated left-orderable groups with only one nontrivial conjugacy class? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.80.
Discussion and literature:
A positive answer to this question implies a positive answer to 21.78. Note that D. Osin constructed torsion-free finitely generated groups with only one nontrivial conjugacy class; see 9.10 in Archive... | 2 | open | Y. Lodha, A. Navas | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,590 | KOU-21.81 | Kourovka Notebook Problem 21.81 | Let $\Gamma$ be a finite simple group and let $N_n(\Gamma)$ denote the set of normal subgroups of the free group $F_n$ of rank $n$ whose quotient is isomorphic to $\Gamma$.
Conjecture: $\operatorname{Aut}(F_n)$ acts transitively on $N_n(\Gamma)$ for $n\geqslant 3$. | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.81.
Discussion and literature:
(J. Wiegold).
This is not true for $n=2$ (B. H. Neumann, H. Neumann, Math. Nachr., 4 (1951), 106--125). A. Lubotzky
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Statu... | 3 | partially_solved | A. Lubotzky | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,591 | KOU-21.82 | Kourovka Notebook Problem 21.82 | Conjecture: For $n\geqslant 3$, there are no finite simple characteristic quotients of the free group $F_n$. | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.82.
Discussion and literature:
This is not true for $n=2$ (W. Y. Chen, A. Lubotzky, P. H. Tiep, to appear in Comment. Math. Helvetici, 2025). A. Lubotzky
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
*... | 3 | partially_solved | A. Lubotzky | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,592 | KOU-21.83 | Kourovka Notebook Problem 21.83 | Conjecture: Metabelian groups are permutation-stable. | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.83.
Discussion and literature:
The function $d_n(\sigma,\tau)=(1/n)\cdot|\{x\in\{1,\ldots,n\}\mid \sigma(x)\ne\tau(x)\}|$ is a distance on the symmetric group $S_n$. For a finitely generated group $G$, an almost-homomorphism is a se... | 3 | open | A. Lubotzky | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,593 | KOU-21.84 | Kourovka Notebook Problem 21.84 | For $\sigma\in S_n$ and $\tau\in S_m$, where $n\leqslant m$, let $d_n^{\mathrm{flex}}(\sigma,\tau)=(1/n)\cdot(|\{x\in\{1,\ldots,n\}\mid \sigma(x)\ne\tau(x)\}|+(m-n))$. An almost-homomorphism $\{f_n\}$ is said to be flexibly close to a homomorphism if there is a sequence of group homomorphisms $\rho_n:G\to S_{m_n}$ with... | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.84.
Discussion and literature:
A. Lubotzky
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** No resolution of the sour... | 2 | open | A. Lubotzky | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,594 | KOU-21.85 | Kourovka Notebook Problem 21.85 | Is a flexibly permutation-stable group always permutation-stable? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.85.
Discussion and literature:
A. Lubotzky
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** No proof that flexible pe... | 2 | open | A. Lubotzky | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,595 | KOU-21.86 | Kourovka Notebook Problem 21.86 | A group $G$ is said to be sofic if for every finite set $F\subseteq G$ containing $1$ and every $\varepsilon>0$ there exist $n\in\mathbb N$ and a map $\phi:F\to S_n$ such that $\phi(1)=1$, $d(\phi(gh),\phi(g)\phi(h))<\varepsilon$ for all $g,h$ such that $gh\in F$, and $\phi(g)$ does not have fixed points for every $g\i... | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.86.
Discussion and literature:
(M. Gromov, B. Weiss).
A. Lubotzky
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** W... | 3 | solved | A. Lubotzky | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] No, not every (AI solution by OpenAI: https://cdn.openai.com/pdf/ten-proofs-oai.pdf). More examples were... | null | null | null | [Discussion #4 literature review, 2026-10-14] **solved (no, not every group is sofic)** and `OWR-5154-008` ("Does there exist a non-sofic group?") β **solved (yes)**; the dataset currently labels both open, while it already labels `ALG-033` and `GREEN-100` (is every group sofic?) solved on the same evidence. Notebook v... | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,596 | KOU-21.87 | Kourovka Notebook Problem 21.87 | Assume that a finite group G has a family of d-generator subgroups whose indices have no common divisor. Is it true that G can be generated by d+1 elements? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.87.
Discussion and literature:
The answer is positive if G is solvable (L. G. Kov\'acs, H.-S. Sim, Indag. Math., 2 (1991), 229--232). For an arbitrary finite group G it is proved that G can be generated by d + 2 elements (A. Lucchin... | 2 | solved | A. Lucchini | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] Yes, it is true (AI solutions confirmed by A. Lucchini: \\ J. DeCaro,\n*Preprint*, July 2026,\nhttps://k... | null | null | null | null | null | The required d+1-generator bound holds. The maintained notebook marks this problem solved. Some cited solutions are preprints or letters; notebook acceptance does not imply journal peer review.
**What remains.** No unresolved part of the stated question is identified in the cited result. | [
{
"url": "https://arxiv.org/abs/1401.0300v46",
"label": "The Kourovka Notebook, 21st edition, version 46 (2026-09-27)"
},
{
"url": "https://arxiv.org/abs/2608.12432",
"label": "R. Sater, solution cited in the notebook (2026)"
},
{
"url": "https://kourovkanotebookorg.wordpress.com/wp-cont... | 2026-09-27T00:00:00 | The Kourovka Notebook, 21st edition, version 46 (2026-09-27): https://arxiv.org/abs/1401.0300v46
R. Sater, solution cited in the notebook (2026): https://arxiv.org/abs/2608.12432
J. DeCaro, July 2026 preprint linked under Problem 21.87 in the notebook: https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/0... | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,597 | KOU-21.88 | Kourovka Notebook Problem 21.88 | Is there a finite non-abelian group $G$ of odd order, with $k(G)$ conjugacy classes, such that $k(G)/|G|=1/17$? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.88.
Discussion and literature:
D. MacHale
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-09-27)
**Status:** solved
**Classification:** SOLVED-IN-LITERATURE
**Current literature assessment.** No finite nonabel... | 2 | solved | D. MacHale | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] No, there are no such groups (B. Beyer de Ryke, *Preprint*, 2026, https://arxiv.org/abs/2608.03003)."
} | null | null | null | null | null | No finite nonabelian group of odd order has commuting probability 1/17. The maintained notebook marks this problem solved. Some cited solutions are preprints or letters; notebook acceptance does not imply journal peer review.
**What remains.** No unresolved part of the stated question is identified in the cited result... | [
{
"url": "https://arxiv.org/abs/1401.0300v46",
"label": "The Kourovka Notebook, 21st edition, version 46 (2026-09-27)"
},
{
"url": "https://arxiv.org/abs/2608.03003",
"label": "B. Beyer de Ryke, solution cited in the notebook (2026)"
},
{
"url": "https://huggingface.co/datasets/ulamai/Un... | 2026-09-27T00:00:00 | The Kourovka Notebook, 21st edition, version 46 (2026-09-27): https://arxiv.org/abs/1401.0300v46
B. Beyer de Ryke, solution cited in the notebook (2026): https://arxiv.org/abs/2608.03003 | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,598 | KOU-21.89 | Kourovka Notebook Problem 21.89 | For $n>39$, is it true that the number of conjugacy classes in the symmetric group $S_n$ of degree $n$ is never a divisor of the order of $S_n$? In other words, is it true that, for $n>39$, the number $p(n)$ of integer partitions of $n$ is never a divisor of $n!$? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.89.
Discussion and literature:
D. MacHale
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** No proof that p(n) never d... | 2 | open | D. MacHale | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,599 | KOU-21.90 | Kourovka Notebook Problem 21.90 | Let $\Gamma$ be a graph of diameter $d$. For $i\in\{1,2,\ldots,d\}$, let $\Gamma_i$ be the graph on the same vertex set as $\Gamma$ with vertices $u,w$ adjacent in $\Gamma_i$ if and only if $d_\Gamma(u,w)=i$. Does there exist a $Q$-polynomial distance-regular graph $\Gamma$ of diameter 3 such that $\Gamma_2$ and $\Gamm... | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.90.
Discussion and literature:
A. A. Makhn\"ev
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** No Q-polynomial dista... | 2 | open | A. A. Makhn\"ev | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,600 | KOU-21.91 | Kourovka Notebook Problem 21.91 | Conjecture: The sum of squares of the degrees of the irreducible $p$-Brauer characters of a finite group $G$ is at least the $p'$-part of $|G|$. | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.91.
Discussion and literature:
(W. Willems).
This is known to be true for $p=2$ (G. Malle, Adv. Math., 380 (2021), Paper no. 107609, 15 pp.). G. Malle
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**S... | 3 | partially_solved | G. Malle | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,601 | KOU-21.92 | Kourovka Notebook Problem 21.92 | Conjecture: The number of irreducible $p$-Brauer characters of a finite group $G$ is bounded above by the maximum of the number of conjugacy classes $k(H)$ in $p'$-subgroups $H$ of $G$. | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.92.
Discussion and literature:
G. Malle, G. Navarro, G. Robinson
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** No ... | 3 | open | G. Malle, G. Navarro, G. Robinson | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,602 | KOU-21.93 | Kourovka Notebook Problem 21.93 | Let $G$ be a group and let $k\geqslant 2$. Let $H_1,\ldots,H_k$ be subgroups of $G$, and $g_1,\ldots,g_k$ elements of $G$ such that the cosets $g_1H_1,\ldots,g_kH_k$ form a partition of $G$. Is it true that $|G:H_i|=|G:H_j|$ for some $i\ne j$? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.93.
Discussion and literature:
(M. Herzog, J. Sch\"onheim).
This is known to be true for groups with a Sylow tower (M. A. Berger, A. Felzenbaum, A. Fraenkel, Fund. Math., 128, no. 3 (1987), 139--144). Also cf. 20.99. L. Margolis
<... | 2 | partially_solved | L. Margolis | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,603 | KOU-21.94 | Kourovka Notebook Problem 21.94 | The Gruenberg--Kegel graph (or the prime graph) GK(G) of a finite group G is a labelled graph with vertex set consisting of all prime divisors of the order of G in which different vertices p and q are adjacent if and only if G contains an element of order pq. Let GK(G) denote the abstract graph obtained from GK(G) by r... | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.94.
Discussion and literature:
N. V. Maslova
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** No infinite family reco... | 2 | open | N. V. Maslova | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,604 | KOU-21.95 | Kourovka Notebook Problem 21.95 | Is there an almost simple but not simple group which is recognizable by the isomorphism type of its Gruenberg--Kegel graph? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.95.
Discussion and literature:
Note that if a group G is recognizable by the isomorphism type of its Gruenberg-- Kegel graph, then G is recognizable by its Gruenberg--Kegel graph, and therefore G is almost simple (P. J. Cameron, N. ... | 2 | open | N. V. Maslova | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,605 | KOU-21.96 | Kourovka Notebook Problem 21.96 | Is it true that a periodic group containing an involution is locally finite if the centralizer of every element of even order is locally finite? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.96.
Discussion and literature:
V. D. Mazurov
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** No proof that the speci... | 2 | open | V. D. Mazurov | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,606 | KOU-21.97 | Kourovka Notebook Problem 21.97 | Is it true that for every positive rational number $r$ there exists a finite group $G$ such that $|\operatorname{Aut}(G)|/|G|=r$? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.97.
Discussion and literature:
(M. T\u{a}rn\u{a}uceanu).
A similar question is answered in the positive for graphs, monoids, partial groups, and posets (R. Molinier, Preprint, 2025, https://arxiv.org/abs/2504.21059). It is also kno... | 2 | solved | R. Molinier | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] Yes, it is true (S. Sureaux, *Preprint*, 2026, https://kourovkanotebookorg.wordpress.com/wp-content/uplo... | null | null | null | null | null | Every positive rational is the automorphism-group order divided by the group order for some finite group. The maintained notebook marks this problem solved. Some cited solutions are preprints or letters; notebook acceptance does not imply journal peer review.
**What remains.** No unresolved part of the stated question... | [
{
"url": "https://arxiv.org/abs/1401.0300v46",
"label": "The Kourovka Notebook, 21st edition, version 46 (2026-09-27)"
},
{
"url": "https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/08/21.97-solution.pdf",
"label": "S. Sureaux, solution cited in the notebook (2026)"
},
{
... | 2026-09-27T00:00:00 | The Kourovka Notebook, 21st edition, version 46 (2026-09-27): https://arxiv.org/abs/1401.0300v46
S. Sureaux, solution cited in the notebook (2026): https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/08/21.97-solution.pdf | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,607 | KOU-21.98 | Kourovka Notebook Problem 21.98 | Let w be a multilinear commutator word, and assume that G is a group where the set of w-values is covered by finitely many cyclic subgroups. Is it true that the verbal subgroup w(G) is finite-by-cyclic? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.98.
Discussion and literature:
This is true for lower central words (G. Cutolo, C. Nicotera, J. Algebra, 324, no. 7 (2010), 1616--1624). M. Morigi
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status... | 2 | partially_solved | M. Morigi | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,608 | KOU-21.99 | Kourovka Notebook Problem 21.99 | Conjecture: If $G$ is a transitive permutation group on a finite set $\Omega$, then for any distinct $\alpha,\beta\in\Omega$ there is an element $g\in G$ with $\alpha^g=\beta$ whose number of fixed points is different from 1. | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.99.
Discussion and literature:
P. M\"uller
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_solved
**Classification:** PARTIAL-PROGRESS
**Current literature assessment.** The conj... | 3 | solved | P. M\"uller | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] The conjecture is refuted in (K. Muliarchyk, *Preprint*, 2026, https://kourovkanotebookorg.wordpress.com... | null | null | null | [Discussion #4 literature review, 2026-10-14] **solved (the conjecture is false)**; the dataset currently labels it partially solved. P. MΓΌller: in a transitive permutation group, for distinct Ξ±, Ξ² there is an element mapping Ξ± to Ξ² whose number of fixed points is not 1. Notebook v48: β21.99, "The conjecture is refuted... | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,609 | KOU-21.100 | Kourovka Notebook Problem 21.100 | Suppose that $A$ and $G$ are finite groups such that $A$ acts coprimely on $G$ by automorphisms. Let $C=C_G(A)$ be the fixed-point subgroup, and let $C'$ denote its derived subgroup. Is it true that the number of $A$-invariant irreducible characters $\chi$ of $G$ whose restriction $\chi_C$ is never zero is exactly $|C/... | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.100.
Discussion and literature:
This would follow if one could show that $\chi_C$ is never zero if and only if the Glauberman--Isaacs correspondent $\chi^*$ of $\chi$ is linear. G. Navarro
<!-- LITERATURE-TRIAGE:BEGIN -->
## Litera... | 2 | open | G. Navarro | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,610 | KOU-21.101 | Kourovka Notebook Problem 21.101 | Which finite almost simple groups are the automorphism groups of regular polytopes of rank 3? In other words, which finite almost simple groups are generated by three involutions two of which commute? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.101.
Discussion and literature:
This question has been answered for finite simple groups; see 7.30 in Archive. Ya. N. Nuzhin
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_solved ... | 2 | partially_solved | Ya. N. Nuzhin | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,611 | KOU-21.102 | Kourovka Notebook Problem 21.102 | Let $V$ be a variety generated by a finite group, and let $f(n)$ be the order of the free group in $V$ on $n$ generators. Is it true that the sequence $\sqrt[n]{\log f(n)}$ has a limit as $n\to\infty$, and this limit is an integer? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.102.
Discussion and literature:
A. Yu. Olshanskii
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_solved
**Classification:** PARTIAL-PROGRESS
**Current literature assessment.** A... | 2 | partially_solved | A. Yu. Olshanskii | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,612 | KOU-21.103 | Kourovka Notebook Problem 21.103 | A Hausdorff topological group G is called minimal if it does not admit a strictly coarser Hausdorff group topology. A topological group is called Raikov complete if its two-sided uniform structure is complete. Is it true that an arbitrary Cartesian product of Raikov complete minimal topological groups remains minimal? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.103.
Discussion and literature:
(V. V. Uspenskii).
It is known that a finite direct product of Raikov complete minimal topological groups is again minimal.
It is known that an arbitrary Cartesian product of centre-free minimal top... | 2 | partially_solved | D. Peng | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,613 | KOU-21.104 | Kourovka Notebook Problem 21.104 | For a group word $w(x_1,\ldots,x_n)$ on $n$ letters, define $e_0(x_1,\ldots,x_n)=x_1$ and $e_{k+1}(x_1,\ldots,x_n)=w(e_k(x_1,\ldots,x_n),\ldots,x_n)$ for all $k\in\mathbb N$. A group $G$ is said to satisfy the Engel type iterated identity $w$ if for all $x_1,\ldots,x_n\in G$ there exists $m\in\mathbb N$ such that $e_m(... | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.104.
Discussion and literature:
This is true in the case of the commutator word $w=[x_1,x_2]$ (G. Fern\'andez-Alcober, M. Noce, G. Tracey, J. Algebra, 554 (2020), 54--77). M. Petschick
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature... | 3 | partially_solved | M. Petschick | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,614 | KOU-21.105 | Kourovka Notebook Problem 21.105 | A group word $w$ is said to be concise in a class $\mathcal C$ of groups if for every group $G$ in $\mathcal C$ such that the set $G_w$ of word values of $w$ in $G$ is finite, the verbal subgroup $w(G)=\langle G_w\rangle$ is also finite. Is every word concise in the class of residually finite groups? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.105.
Discussion and literature:
(D. Segal).
See also Archive 2.45.
See (D. Segal, Words. Notes on verbal width in groups, Cambridge Univ. Press, 2009) for a partial solution. M. Petschick
<!-- LITERATURE-TRIAGE:BEGIN -->
## Liter... | 2 | partially_solved | M. Petschick | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,615 | KOU-21.106 | Kourovka Notebook Problem 21.106 | A first order formula $\phi(x)$ in the group language with one free variable is said to be concise in a class $\mathcal C$ of groups if for every group $G$ in $\mathcal C$ such that the set $G_\phi$ of elements in $G$ satisfying $\phi$ is finite, the subgroup $\phi(G)$ generated by $G_\phi$ is also finite. Is every for... | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.106.
Discussion and literature:
Cf. 21.105 and Archive 2.45. M. Petschick
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_solved
**Classification:** PARTIAL-PROGRESS
**Current li... | 2 | solved | M. Petschick | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] No, not every (AI solution confirmed by M. Petschick: \\ Achyuth Jayadevan, *Preprint of 12 September 20... | null | null | null | [Discussion #4 literature review, 2026-10-14] **solved (no)**; the dataset currently labels it partially solved. M. Petschick: is every first-order formula with one free variable concise in the class of residually finite groups? Notebook v48: β21.106, "No, not every", AI solution confirmed by M. Petschick (A. Jayadevan... | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,616 | KOU-21.107 | Kourovka Notebook Problem 21.107 | A sequence $\{F_n\}$ of pairwise disjoint finite subsets of a topological group is called expansive if for every open subset $U$ there is a number $m$ such that $F_n\cap U\ne\emptyset$ for all $n>m$. Suppose that a countable group $G$ can be partitioned into countably many dense subsets. Is it true that in $G$ there ex... | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.107.
Discussion and literature:
Cf. 15.80. I. V. Protasov
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** The earlie... | 2 | open | I. V. Protasov | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,617 | KOU-21.108 | Kourovka Notebook Problem 21.108 | For a finite group $G$ let $\operatorname{Cod}(G)$ denote the set of irreducible character codegrees of $G$ (see 20.78). Define $\sigma(G)=\max\{|\pi(m)|:m\in\operatorname{Cod}(G)\}$, where $\pi(m)$ denotes the set of prime divisors of an integer $m$. Can the constant $k$ be taken as 4? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.108.
Discussion and literature:
It is proved that there exists a constant $k$ such that $|\pi(G)|\leqslant k\cdot\sigma(G)$ for every finite group $G$ (Y. Yang, G. Qian, J. Algebra, 478 (2017), 215--219), but the estimate provided f... | 2 | partially_solved | G. Qian | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,618 | KOU-21.109 | Kourovka Notebook Problem 21.109 | Conjecture: The derived length of a finite solvable group $G$ does not exceed $|\operatorname{Cod}(G)|-1$. | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.109.
Discussion and literature:
The Fitting height of $G$ is known to be at most $\min\{|\operatorname{Cod}(G)|-1,\ |\operatorname{Cod}(G)|/2+1\}$ (G. Qian, Y. Zeng, J. Group Theory, to appear). G. Qian
<!-- LITERATURE-TRIAGE:BEGIN... | 3 | partially_solved | G. Qian | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,619 | KOU-21.110 | Kourovka Notebook Problem 21.110 | Let $S$ be a nonabelian finite simple group, and $x$ a nonidentity automorphism of $S$. Let $\alpha(x)$ be the smallest number of conjugates of $x$ in $G=\langle x,\operatorname{Inn}S\rangle$ that generate $G$.
(a) Conjecture: If $S$ is an exceptional group of Lie type, then $\alpha(x)\leqslant 5$ for every nonidentit... | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.110.
Discussion and literature:
The values of $\alpha(x)$ had been studied in (R. Guralnick, J. Saxl, J. Algebra, 268, no. 2 (2003), 519--571).
Part (a) is attributed to R. Guralnick and J. Saxl. D. O. Revin
<!-- LITERATURE-TRIAGE... | 3 | open | D. O. Revin | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,620 | KOU-21.111 | Kourovka Notebook Problem 21.111 | Let $S$ be a finite simple nonabelian group that is not isomorphic to any group ${}^2B_2(q)$. A nonidentity automorphism $x$ of $S$ is called a $\tau$-automorphism if every two conjugates of $x$ in $\langle x,\operatorname{Inn}(S)\rangle$ generate a subgroup of order not divisible by 3. If $S$ admits a $\tau$-automorph... | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.111.
Discussion and literature:
D. O. Revin, N. Y. Yang
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** No complete ... | 2 | open | D. O. Revin, N. Y. Yang | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,621 | KOU-21.112 | Kourovka Notebook Problem 21.112 | A nonempty class $\mathcal X$ of finite groups is said to be complete if $\mathcal X$ is closed under taking subgroups, homomorphic images, and extensions. The symmetric boundary of a complete class $\mathcal X$ other than the class of all finite groups is defined as the largest integer $n$ such that $S_n\in\mathcal X$... | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.112.
Discussion and literature:
Every positive integer $n\ne 3$ coincides with the symmetric boundary of some complete class. It is proved (mod CFSG, D. O. Revin, Algebra i Analiz, 37, no. 1 (2025), 141--176 (Russian)) that, for eve... | 2 | partially_solved | D. O. Revin, N. Y. Yang | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,622 | KOU-21.113 | Kourovka Notebook Problem 21.113 | Let $G$ be a finite group and $p$ be a prime. Let $\Psi_{p,G}$ be the class function of $G$ which vanishes on all $p$-singular elements of $G$ and whose value at each $p$-regular element $x$ of $G$ is the number of $p$-elements of $C_G(x)$.
(a) Is it true that $\Psi_{p,G}$ is a character of $G$?
(b) If yes, can $\Psi... | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.113.
Discussion and literature:
It is known that $\Psi_{p,G}$ is a character when $G\cong S_n$ for any positive integer $n$ and any prime $p$ (T. Scharf, J. Algebra, 139, no. 2 (1991), 446--457). G. Robinson
<!-- LITERATURE-TRIAGE:... | 2 | partially_solved | G. Robinson | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,623 | KOU-21.114 | Kourovka Notebook Problem 21.114 | A finite group G is called weakly ab-maximal if |H : [H, H]| $\leqslant$ |G : [G, G]| for all H $\leqslant$ G. Do weakly ab-maximal groups have bounded derived length? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.114.
Discussion and literature:
It is known that weakly ab-maximal groups are direct products of weakly ab-maximal p-groups (F. Lisi, L. Sabatini, J. Group Theory, 27 (2024), 1203--1217). L. Sabatini
<!-- LITERATURE-TRIAGE:BEGIN --... | 2 | partially_solved | L. Sabatini | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,624 | KOU-21.115 | Kourovka Notebook Problem 21.115 | Let $C_1,\ldots,C_n$ be (left or right) cosets of a finite group $G$ such that $U:=C_1\cup\cdots\cup C_n$ is not $G$. Is it always true that $|G\setminus U|\geqslant |G|/2^n$? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.115.
Discussion and literature:
Affirmative answers are known in some special cases (B. Sambale, M. T\u{a}rn\u{a}uceanu, J. Algebraic Combin., 55 (2022), 979--987). B. Sambale
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (... | 2 | solved | B. Sambale | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-09-28T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] Yes, it is true (Anonymous AI-proof confirmed by B. Sambale: *Preprint*, August 2026, https://www.unsolv... | null | null | null | null | null | Sambaleβs September 2026 unrefereed preprint proves that the complement $A$ of a proper union of $n$ left cosets covers the group with at most $2^n$ left translates. For finite $G$, this gives $|A|\ge |G|/2^n$. Right cosets are left cosets of conjugate subgroups, so mixed left/right cosets are covered too.
**What rema... | [
{
"url": "https://arxiv.org/abs/2609.09052",
"label": "B. Sambale, Coset covers and complements β September 2026 preprint"
},
{
"url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"label": "UnsolvedMath discussion #4 β Alper Ferudun, status updates and follow-ups"
}
] | 2026-09-28T00:00:00 | B. Sambale, Coset covers and complements β September 2026 preprint: https://arxiv.org/abs/2609.09052 | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,625 | KOU-21.116 | Kourovka Notebook Problem 21.116 | A group is boundedly acyclic if its bounded cohomology with trivial real coefficients vanishes in all positive degrees. Is every branch group boundedly acyclic? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.116.
Discussion and literature:
E. Schesler
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** No proof or counterexamp... | 2 | open | E. Schesler | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,626 | KOU-21.117 | Kourovka Notebook Problem 21.117 | (a) Does there exist a finitely generated simple group that is of exponential growth but not of uniformly exponential growth?
(b) Does there exist a finitely generated hereditarily just-infinite group that is of exponential growth but not of uniformly exponential growth? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.117.
Discussion and literature:
E. Schesler
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-09-27)
**Status:** solved
**Classification:** SOLVED-IN-LITERATURE
**Current literature assessment.** Thompsonβs grou... | 2 | solved | E. Schesler | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] Yes, Thompson's group $V$ provides an affirmative answer to both questions (R. Sauer, E. Schesler, *Prep... | null | null | null | null | null | Thompsonβs group V answers both existence questions affirmatively. The maintained notebook marks this problem solved. Some cited solutions are preprints or letters; notebook acceptance does not imply journal peer review.
**What remains.** No unresolved part of the stated question is identified in the cited result. | [
{
"url": "https://arxiv.org/abs/1401.0300v46",
"label": "The Kourovka Notebook, 21st edition, version 46 (2026-09-27)"
},
{
"url": "https://arxiv.org/abs/2605.30163",
"label": "R. Sauer and E. Schesler, solution cited in the notebook (2026)"
},
{
"url": "https://huggingface.co/datasets/u... | 2026-09-27T00:00:00 | The Kourovka Notebook, 21st edition, version 46 (2026-09-27): https://arxiv.org/abs/1401.0300v46
R. Sauer and E. Schesler, solution cited in the notebook (2026): https://arxiv.org/abs/2605.30163 | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,627 | KOU-21.118 | Kourovka Notebook Problem 21.118 | Is there any group which is not isomorphic to the quotient of a residually finite group by an amenable normal subgroup? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.118.
Discussion and literature:
(A. Thom).
E. Schesler
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** No group out... | 2 | open | E. Schesler | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,628 | KOU-21.119 | Kourovka Notebook Problem 21.119 | Does there exist a group $G$ that contains a family $(G_n)_{n\in\mathbb N}$ of finite-index subgroups such that for every $n$ there is a homomorphism $f_n:G_n\to\mathbb Z$ whose kernel is of type $F_n$, but not of type $F_{n+1}$? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.119.
Discussion and literature:
E. Schesler
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** No single group with the... | 2 | open | E. Schesler | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,629 | KOU-21.120 | Kourovka Notebook Problem 21.120 | A pro-p group is (relatively) strictly finitely presented if it is the pro-p completion of a group that is finitely presented (respectively, finitely presented in some finitely-based variety of groups). A pro-p group is finitely axiomatizable if it is determined up to isomorphism by a single sentence in the first-order... | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.120.
Discussion and literature:
See (A. Nies, K. Tent, D. Segal, Proc. London Math. Soc. (3), 123 (2021), 597-- 635; D. Segal, Preprint, 2025, https://arxiv.org/abs/2505.04816). D. Segal
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literatu... | 2 | partially_solved | D. Segal | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,630 | KOU-21.121 | Kourovka Notebook Problem 21.121 | Let $p$ be a prime number. A group $\Gamma$ is called $p$-Jordan if there exist constants $J$ and $e$ such that any finite subgroup $G\subset\Gamma$ contains a normal abelian subgroup of order coprime to $p$ and of index at most $J\cdot |G(p)|^e$. (For example by the results of Brauer--Feit and Larsen--Pink, for any fi... | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.121.
Discussion and literature:
C. Shramov
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_solved
**Classification:** PARTIAL-PROGRESS
**Current literature assessment.** Importan... | 2 | partially_solved | C. Shramov | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "partially_solved",
"checked_at": "2026-10-16",
"notes": "Subproblem confirmed solved (*) in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] No, not always (AI solution confirmed by C. Shramov: \\ Achyuth Jayadevan, *Preprint of 14 Sep... | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,631 | KOU-21.122 | Kourovka Notebook Problem 21.122 | Let w be a group word, and G a profinite group. Is it true that the cardinality of the set of w-values in G is either finite or at least continuum? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.122.
Discussion and literature:
An affirmative answer is known for several important words; see (E. Detomi, B. Klopsch, P. Shumyatsky, J. London Math. Soc. (2), 102 (2020), 977--993). P. Shumyatsky
<!-- LITERATURE-TRIAGE:BEGIN -->
... | 2 | partially_solved | P. Shumyatsky | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,632 | KOU-21.123 | Kourovka Notebook Problem 21.123 | Is it true that the extension of the A. Agrachev--R. Gamkrelidze construction of groups from pre-Lie rings suggested in Definition 66 produces groups from pre-Lie rings? If this extended construction does give a group, then it also gives a brace, and so an affirmative answer to this question would have consequences for... | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.123.
Discussion and literature:
A. Agrachev--R. Gamkrelidze construction: (J. Soviet Math., 17 (1981), 1650--1675). Definition 66: (A. Smoktunowicz, J. Pure Appl. Algebra, 229, no. 12 (2025), 108128). A. Smoktunowicz
<!-- LITERATUR... | 2 | open | A. Smoktunowicz | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,633 | KOU-21.124 | Kourovka Notebook Problem 21.124 | A group G is said to be virtually special if G has a finite-index subgroup isomorphic to the fundamental group of a special complex. A group G is called a CAT(0) group if it acts properly discontinuously and cocompactly by isometries on a CAT(0) metric space.
a) Is every CAT(0) free-by-cyclic group virtually special?
... | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.124.
Discussion and literature:
Special complex is meant in the sense of F. Haglund, D. T. Wise, Geom. Funct. Anal., 17, no. 5 (2008), 1551--1620; cf 20.60. I. Soroko
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2... | 2 | open | I. Soroko | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,634 | KOU-21.125 | Kourovka Notebook Problem 21.125 | Let $F_m$ be a free group of rank $m$ and let $\varphi\in\operatorname{Aut}(F_m)$ be a polynomially growing automorphism of maximal degree $m-1$, which means that for some (equivalently, any) free basis $\{x_1,\ldots,x_m\}$ of $F_m$, the sequence $\max_i |\varphi^n(x_i)|$ grows at the rate of $n^{m-1}$, where $|g|$ den... | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.125.
Discussion and literature:
(M. Bridson).
I. Soroko
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** No virtual-... | 2 | open | I. Soroko | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,635 | KOU-21.126 | Kourovka Notebook Problem 21.126 | Do there exist finitely presented subgroups of right-angled Artin groups whose Dehn functions are super-exponential, or sub-exponential but not polynomial? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.126.
Discussion and literature:
(N. Brady).
Such subgroups are known to exist in general CAT(0) groups, whereas the only Dehn functions currently realized for subgroups of right-angled Artin groups are exponential and polynomial of... | 2 | partially_solved | I. Soroko | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,636 | KOU-21.127 | Kourovka Notebook Problem 21.127 | Let G be a right-angled Artin group. Is the stable commutator length scl(g) a rational number for every g $\in$ [G, G]? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.127.
Discussion and literature:
For free groups this is true by Calegari's Rationality Theorem. (See 18.40 for the definition of scl(g).) I. Soroko
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Statu... | 2 | partially_solved | I. Soroko | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,637 | KOU-21.128 | Kourovka Notebook Problem 21.128 | Two groups $G_1$ and $G_2$ are said to be commensurable if there exist finite-index subgroups $H_1\leqslant G_1$ and $H_2\leqslant G_2$ (not necessarily of the same index) such that $H_1\cong H_2$. Let $A[F_4]$ and $A[H_4]$ denote the Artin groups of spherical types $F_4$ and $H_4$, respectively. Are these two groups c... | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.128.
Discussion and literature:
This is the most difficult case in the classification of Artin groups of spherical type up to commensurability. I. Soroko
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
*... | 2 | open | I. Soroko | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,638 | KOU-21.129 | Kourovka Notebook Problem 21.129 | If two Artin groups of spherical type are quasi-isometric, must they be commensurable? (This is not true for right-angled Artin groups.) | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.129.
Discussion and literature:
I. Soroko
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** No proof that quasi-isomet... | 2 | open | I. Soroko | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,639 | KOU-21.130 | Kourovka Notebook Problem 21.130 | Conjecture: Let $G$ be a finite additive abelian group with $|G|$ odd. Then any subset $A$ of $G$ with $|A|=n>2$ can be written as $\{a_1,\ldots,a_n\}$ in such a way that all the sums $a_1+a_2,a_2+a_3,\ldots,a_{n-1}+a_n,a_n+a_1$ are distinct. | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.130.
Discussion and literature:
Z. W. Sun
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** No proof or counterexample... | 3 | open | Z. W. Sun | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,640 | KOU-21.131 | Kourovka Notebook Problem 21.131 | Construct a homomorphism of a subgroup of a Golod group onto an infinite AT-group. | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.131.
Discussion and literature:
Golod group: see 9.76. AT-group: (A. V. Rozhkov, Math. Notes, 40, no. 5 (1986), 827--836). Cf. 13.55 A. V. Timofeenko
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Sta... | 2 | open | A. V. Timofeenko | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,641 | KOU-21.132 | Kourovka Notebook Problem 21.132 | Based on the development of E. S. Golod's construction, for each prime number p, construct a finitely generated residually finite p-group with a non-trivial finite centre. | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.132.
Discussion and literature:
Development of E. S. Golod's construction: see, for example, Discrete Math. Appl., 23, no. 5--6 (2013), 491--501.
Such groups with infinite and trivial centres are known (see 9.76 and Archive 11.101)... | 2 | open | A. V. Timofeenko | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,642 | KOU-21.133 | Kourovka Notebook Problem 21.133 | Does a group need to have a subnormal abelian series if every countable subgroup of it has such a series? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.133.
Discussion and literature:
M. Trombetti
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** No local-to-global theo... | 2 | open | M. Trombetti | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,643 | KOU-21.134 | Kourovka Notebook Problem 21.134 | For a finite group $G$, let the type of $G$ be the function on positive integers whose value at $n$ is the number of solutions of the equation $x^n=1$ in $G$.
a) Is it true that a group having the same type as a group with trivial solvable radical must also have trivial solvable radical?
b) Is it true that a group ha... | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.134.
Discussion and literature:
Note that there are solvable and nonsolvable groups with the same type (see 12.37).
This is true for a group having the same type as a simple group, as follows from the affirmative answer to 12.39 (i... | 2 | solved | A. V. Vasil'ev | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] It turned out that a negative answer to both questions had already been given by J. G. Thompson; see, fo... | null | null | null | null | null | Both questions have negative answers already due to J. G. Thompson, as explained by Y. Li and W. Shi, Ricerche di Matematica 74 (2025), 559β563. The maintained notebook marks this problem solved. Some cited solutions are preprints or letters; notebook acceptance does not imply journal peer review.
**What remains.** No... | [
{
"url": "https://arxiv.org/abs/1401.0300v46",
"label": "The Kourovka Notebook, 21st edition, version 46 (2026-09-27)"
},
{
"url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"label": "UnsolvedMath discussion #4 β Alper Ferudun, status updates and follow-ups"
}
] | 2026-09-27T00:00:00 | The Kourovka Notebook, 21st edition, version 46 (2026-09-27): https://arxiv.org/abs/1401.0300v46 | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,644 | KOU-21.135 | Kourovka Notebook Problem 21.135 | For a finite group $G$, let $\chi_1(G)$ denote the totality of the degrees of all irreducible complex characters of $G$ with allowance for their multiplicities. Suppose that $H$ is a finite group with $\chi_1(H)=\chi_1(G)$. If $G$ has trivial solvable radical, must $H$ also have trivial solvable radical? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.135.
Discussion and literature:
A. V. Vasil'ev
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** No theorem that full ... | 2 | open | A. V. Vasil'ev | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,645 | KOU-21.136 | Kourovka Notebook Problem 21.136 | Let G be a profinite group with fewer than $2^{\aleph_0}$ conjugacy classes of elements of infinite order. Must G be a torsion group? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.136.
Discussion and literature:
This holds in the case when G is finitely generated (J. S. Wilson, Arch. Math., 120 (2023), 557--563). John S. Wilson
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Sta... | 2 | partially_solved | John S. Wilson | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,646 | KOU-21.137 | Kourovka Notebook Problem 21.137 | If the $p$-th powers in a finite $p$-group form a subgroup, must that subgroup be powerful? That is, for $p\ne 2$, if the $p$-th powers in a $p$-group of exponent $p^2$ form a subgroup, must that subgroup be abelian? For a 2-group of exponent 8, if the squares form a subgroup, must that subgroup be abelian? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.137.
Discussion and literature:
L. Wilson
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-09-27)
**Status:** solved
**Classification:** SOLVED-IN-LITERATURE
**Current literature assessment.** Counterexamples e... | 2 | solved | L. Wilson | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] No, it need not be. For $p=3$ a counterexample is constructed in (K. Muliarchyk, *Preprint*, 2026,\nhttp... | null | null | null | null | null | Counterexamples exist for p=3 (K. Muliarchyk) and p=2 (A. Chang, letter of 1 August 2026); the latter is the Sylow 2-subgroup of S_8. The maintained notebook marks this problem solved. Some cited solutions are preprints or letters; notebook acceptance does not imply journal peer review.
**What remains.** No unresolved... | [
{
"url": "https://arxiv.org/abs/1401.0300v46",
"label": "The Kourovka Notebook, 21st edition, version 46 (2026-09-27)"
},
{
"url": "https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/08/kourovka_21_137_counterexample.pdf",
"label": "K. Muliarchyk, solution cited in the notebook (2... | 2026-09-27T00:00:00 | The Kourovka Notebook, 21st edition, version 46 (2026-09-27): https://arxiv.org/abs/1401.0300v46
K. Muliarchyk, solution cited in the notebook (2026): https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/08/kourovka_21_137_counterexample.pdf | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,647 | KOU-21.138 | Kourovka Notebook Problem 21.138 | Let G be an infinite finitely presented group such that every subgroup of infinite index is free. Must G be isomorphic to either a free group or a surface group? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.138.
Discussion and literature:
H. Wilton
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** No classification of finit... | 2 | open | H. Wilton | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,648 | KOU-21.139 | Kourovka Notebook Problem 21.139 | Let G be a hyperbolic group which is virtually compact special in the sense of Haglund--Wise. Suppose that the set of second Betti numbers of the finite-index subgroups of G is bounded. Must G be virtually either a free group or a surface group? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.139.
Discussion and literature:
H. Wilton
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** No virtual free-or-surface... | 2 | open | H. Wilton | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,649 | KOU-21.140 | Kourovka Notebook Problem 21.140 | Let G be a torsion-free group of type $F_\infty$ of infinite cohomological dimension. Must G contain a copy of Thompson's group F? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.140.
Discussion and literature:
S. Witzel, M. C. B. Zaremsky
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** No theo... | 2 | open | S. Witzel, M. C. B. Zaremsky | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,650 | KOU-21.141 | Kourovka Notebook Problem 21.141 | Let $G=G_1\amalg_H G_2$ be a free pro-$p$ product of coherent pro-$p$ groups with polycyclic amalgamation. Is $G$ coherent? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.141.
Discussion and literature:
For abstract groups this is known to be true. A group is said to be coherent if each of its finitely generated subgroups is finitely presented, and in the question the coherency is used in the pro-$p$... | 2 | partially_solved | P. A. Zalesskii | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,651 | KOU-21.142 | Kourovka Notebook Problem 21.142 | A group $G$ is said to be invariably generated by $a$ and $b$ if $G$ is generated by the conjugates $a^g,b^h$ for every $g,h$. Let $p\ne q$ be fixed primes. Does every finite group embed into a finite group invariably generated by an element of order $p$ and an element of order $q$? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.142.
Discussion and literature:
P. A. Zalesskii
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-09-27)
**Status:** solved
**Classification:** SOLVED-IN-LITERATURE
**Current literature assessment.** For any dis... | 2 | solved | P. A. Zalesskii | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] No, for any two primes $p\\neq q$ there is a finite group that does not embed into a\nfinite group invar... | null | null | null | null | null | For any distinct primes p and q there is a finite group that admits no embedding of the required kind. The maintained notebook marks this problem solved. Some cited solutions are preprints or letters; notebook acceptance does not imply journal peer review.
**What remains.** No unresolved part of the stated question is... | [
{
"url": "https://arxiv.org/abs/1401.0300v46",
"label": "The Kourovka Notebook, 21st edition, version 46 (2026-09-27)"
},
{
"url": "https://arxiv.org/abs/2608.00703",
"label": "T. Gong, M. R. Zeng and Y. Yang, solution cited in the notebook (2026)"
},
{
"url": "https://arxiv.org/abs/2608... | 2026-09-27T00:00:00 | The Kourovka Notebook, 21st edition, version 46 (2026-09-27): https://arxiv.org/abs/1401.0300v46
T. Gong, M. R. Zeng and Y. Yang, solution cited in the notebook (2026): https://arxiv.org/abs/2608.00703
I. Capdeboscq and C. Parker, independent results on invariable generation (2026): https://arxiv.org/abs/2608.03935 | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,652 | KOU-21.143 | Kourovka Notebook Problem 21.143 | (Well-known problem). Is Thompson's group F automatic? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.143.
Discussion and literature:
M. C. B. Zaremsky
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** Automaticity of Th... | 3 | open | M. C. B. Zaremsky | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,653 | KOU-21.144 | Kourovka Notebook Problem 21.144 | Conjecture: Every subgroup of Thompson's group F is either elementary amenable or else contains a subgroup isomorphic to F. | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.144.
Discussion and literature:
(M. Brin, M. Sapir).
M. C. B. Zaremsky
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment... | 3 | open | M. C. B. Zaremsky | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,654 | KOU-21.145 | Kourovka Notebook Problem 21.145 | Is Thompson's group F quasi-isometric
(a) to F $\times$ Z?
(b) to F $\times$ F? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.145.
Discussion and literature:
(M. Bridson).
M. C. B. Zaremsky
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** No ... | 2 | open | M. C. B. Zaremsky | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,655 | KOU-21.146 | Kourovka Notebook Problem 21.146 | (Well-known problem). A classifying space for a group $G$ is a connected CW-complex with fundamental group $G$ and all higher homotopy groups trivial. A group is of type $F_n$ if it has a classifying space with finite $n$-skeleton. For example, type $F_1$ is equivalent to finite generation, and type $F_2$ is equivalent... | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.146.
Discussion and literature:
M. C. B. Zaremsky
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** No embedding theor... | 3 | open | M. C. B. Zaremsky | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,656 | KOU-21.147 | Kourovka Notebook Problem 21.147 | A subgroup H of a right-orderable group G is said to be right-relatively convex if it is convex under some right ordering on G. Is the lattice of right-relatively convex subgroups of a right-orderable group always a sublattice of the lattice of its subgroups? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.147.
Discussion and literature:
(V. M. Kopytov, N. Ya. Medvedev).
A. V. Zenkov
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-09-27)
**Status:** solved
**Classification:** SOLVED-IN-LITERATURE
**Current lite... | 2 | solved | A. V. Zenkov | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] No, not always (AI solution confirmed by A. V. Zenkov: \\ P. Monticone, *Preprint*, 2026,\nhttps://kouro... | null | null | null | null | null | Right-relatively convex subgroups need not form a sublattice of the subgroup lattice. The maintained notebook marks this problem solved. Some cited solutions are preprints or letters; notebook acceptance does not imply journal peer review.
**What remains.** No unresolved part of the stated question is identified in th... | [
{
"url": "https://arxiv.org/abs/1401.0300v46",
"label": "The Kourovka Notebook, 21st edition, version 46 (2026-09-27)"
},
{
"url": "https://arxiv.org/abs/2607.17477",
"label": "W. van Doorn, E. Judin, P. Monticone and D. Morrison, solution cited in the notebook (2026)"
},
{
"url": "https... | 2026-09-27T00:00:00 | The Kourovka Notebook, 21st edition, version 46 (2026-09-27): https://arxiv.org/abs/1401.0300v46
W. van Doorn, E. Judin, P. Monticone and D. Morrison, solution cited in the notebook (2026): https://arxiv.org/abs/2607.17477 | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,657 | KOU-21.148 | Kourovka Notebook Problem 21.148 | Is it true that the lattice of right-relatively convex subgroups of a right-orderable group is distributive if and only if it is a chain? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.148.
Discussion and literature:
(V. M. Kopytov, N. Ya. Medvedev).
A. V. Zenkov
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature as... | 2 | solved | A. V. Zenkov | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] No, it is not (K. Muliarchyk, *Preprint*, 2026, https://kourovkanotebookorg.wordpress.com/wp-content/upl... | null | null | null | [Discussion #4 literature review, 2026-10-14] **solved (no)**; the dataset currently labels it open. A. V. Zenkov (after V. M. Kopytov, N. Ya. Medvedev): is the lattice of right-relatively convex subgroups of a right-orderable group distributive if and only if it is a chain? Notebook v48: β21.148, "No, it is not", with... | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,658 | KOU-21.149 | Kourovka Notebook Problem 21.149 | Are there order automorphisms of Dlab groups that are not inner automorphisms? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.149.
Discussion and literature:
(V. M. Kopytov, N. Ya. Medvedev).
A. V. Zenkov
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature as... | 2 | open | A. V. Zenkov | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,659 | KOU-21.150 | Kourovka Notebook Problem 21.150 | Let $G$ be an extension of a normal elementary abelian subgroup $A$ by an elementary abelian group $B\cong G/A$ such that $A$ contains an element $a$ with $C_B(a)=1$. Is it true that the rank of the subgroup $Z(\langle a,B\rangle)\cap\langle a,B\rangle'$ is at most the rank of $B$? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.150.
Discussion and literature:
V. I. Zenkov
No, not always. Let $A=\mathbb F_3[x,y]/I^3$ be the additive group of the quotient of the polynomial algebra $\mathbb F_3[x,y]$ by the ideal $I^3$, where $I$ is the ideal generated by $x... | 2 | solved | V. I. Zenkov | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1β21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1β21 (1965β2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] No, not always. Let $A=\\mathbb{F}_3[x, y]/I^3$ be the additive group of the quotient of the polynomial ... | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,660 | KP-1.1 | Kirby Problem 1.1 | Is the crossing number additive under connected sum:
$c(K_{1}\#K_{2}) = c(K_{1}) + c(K_{2})$? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.1.
Literature notes:
This is Problem 1.65 in [Kir97]. Many special cases are known.
The most famous case is due to Kauffman [Kau87], Murasugi [Mur87], and
Thistlewaite [Thi87], who showed the conjecture for alternating knots (and... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | {
"source_url": "https://arxiv.org/abs/2608.13789",
"checked_at": "2026-10-03",
"kind": "partial",
"evidence": "unrefereed_claim"
} | null | null | null | [Discussion #4 literature review, 2026-10-03] keep **partially_solved**; update the literature note. K3 2026, Problem 1.1 (p. 12; scribe I. Zemke; Problem 1.65 of Kirby's 1997 list) asks whether c(Kβ#Kβ) = c(Kβ) + c(Kβ). A. Kawauchi, [arXiv:2608.13789](https://arxiv.org/abs/2608.13789) (Aug. 2026), Thm 1.4, claims addi... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,661 | KP-1.2 | Kirby Problem 1.2 | (a) Show that if $P$ is a nontrivial satellite operator and $K_{P}$ is a nontrivial
satellite of a knot $K$, then
$$
c(K_{P}) \geq c(K),
$$
where $c$ denotes the crossing number.
(b) More generally, show that if $P$ is a satellite operator then
$$
c(K_{P}) \geq c(K) \cdot w(P)^{2},
$$
where $w(P)$ denotes the geome... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.2.
Literature notes:
(1) Part (a) is Problem 1.67 in [Kir97]. Part (b) appears as the Satellite
Crossing Number Conjecture in [BM24a].
(2) Note that a special case of this problem is whether
$$
c(K_{1}\#K_{2}) \geq c(K_{1}),
$$
... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,662 | KP-1.3 | Kirby Problem 1.3 | How does unknotting number behave under connected sum and
mutation?
(a) Does the connected sum of $n$ nontrivial knots have unknotting number at
least $n$?
(b) Is unknotting number invariant under mutation? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.3.
Literature notes:
(1) Problem 1.69 (B) from [Kir97] asks if unknotting number is additive; a
recent preprint [BH25] claims an example where it fails to be additive,
where the knots in question are the $(2, 7)$ torus knot and i... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,663 | KP-1.4 | Kirby Problem 1.4 | Let $P$ be a nontrivial satellite pattern with winding number
$w(P) \neq 0$. Then for any nontrivial knot $K$ and its satellite $K_{P}$ , one has
$$
u(K_{P}) \geq w(P) + 1.
$$ | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.4.
Literature notes:
(1) This problem appears as [HLP22, Conjecture 1.10].
(2) One might guess that $w(P)^{2}$ is a lower bound. However, it can be shown
that
$$
u((T_{2,3})_{2,1}) \leq 3,
$$
where $K_{p,q}$ denotes the $(p, q)... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.