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Theorems that help decompose a finite group based on prime factors of its order
In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow that give detailed information about the number of subgroups of fix... | 27233 | abstract_algebra |
Group of even permutations of a finite set
In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of n elements is called the alternating group of degree n, or the alternating group on n letters and denoted by A"n" or Alt("n").
Basic properties.
For "n" &g... | 25195 | abstract_algebra |
Type of group in abstract algebra
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group $\mathrm{S}_n$ defined over a finite set of $n$ ... | 14315 | abstract_algebra |
Cardinality of a mathematical group, or of the subgroup generated by an element
In mathematics, the order of a finite group is the number of its elements. If a group is not finite, one says that its order is "infinite". The "order" of an element of a group (also called period length or period) is the order of the subgr... | 199412 | abstract_algebra |
In group theory, a branch of mathematics, the automorphisms and outer automorphisms of the symmetric groups and alternating groups are both standard examples of these automorphisms, and objects of study in their own right, particularly the exceptional outer automorphism of S6, the symmetric group on 6 elements.
Formall... | 1836466 | abstract_algebra |
Group in which the order of every element is a power of p
In mathematics, specifically group theory, given a prime number "p", a "p"-group is a group in which the order of every element is a power of "p". That is, for each element "g" of a "p"-group "G", there exists a nonnegative integer "n" such that the product of "... | 12162 | abstract_algebra |
Mathematical group that can be generated as the set of powers of a single element
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C"n", that is generated by a single element. That is, it is a set of invertible elements with a single associative b... | 26391 | abstract_algebra |
The order of a subgroup of a finite group G divides the order of G
In the mathematical field of group theory, Lagrange's theorem is a theorem that states that for any finite group G, the order (number of elements) of every subgroup of G divides the order of G. The theorem is named after Joseph-Louis Lagrange. The follo... | 15371 | abstract_algebra |
Representation of groups by permutations
In group theory, Cayley's theorem, named in honour of Arthur Cayley, states that every group G is isomorphic to a subgroup of a symmetric group.
More specifically, G is isomorphic to a subgroup of the symmetric group $\operatorname{Sym}(G)$ whose elements are the permutations o... | 50326 | abstract_algebra |
Group that can be constructed from abelian groups using extensions
In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminates in the t... | 49669 | abstract_algebra |
Commutative group where every element is the sum of elements from one finite subset
In abstract algebra, an abelian group $(G,+)$ is called finitely generated if there exist finitely many elements $x_1,\dots,x_s$ in $G$ such that every $x$ in $G$ can be written in the form $x = n_1x_1 + n_2x_2 + \cdots + n_sx_s$ for so... | 29251 | abstract_algebra |
In mathematics, two Prüfer theorems, named after Heinz Prüfer, describe the structure of certain infinite abelian groups. They have been generalized by L. Ya. Kulikov.
Statement.
Let "A" be an abelian group. If "A" is finitely generated then by the fundamental theorem of finitely generated abelian groups, "A" is decomp... | 3117305 | abstract_algebra |
In mathematics, in the field of group theory, a quasinormal subgroup, or permutable subgroup, is a subgroup of a group that commutes (permutes) with every other subgroup with respect to the product of subgroups. The term "quasinormal subgroup" was introduced by Øystein Ore in 1937.
Two subgroups are said to permute (or... | 667154 | abstract_algebra |
Existence of group elements of prime order
In mathematics, specifically group theory, Cauchy's theorem states that if G is a finite group and p is a prime number dividing the order of G (the number of elements in G), then G contains an element of order p. That is, there is x in G such that p is the smallest positive in... | 677473 | abstract_algebra |
In abstract algebra, the focal subgroup theorem describes the fusion of elements in a Sylow subgroup of a finite group. The focal subgroup theorem was introduced in and is the "first major application of the transfer" according to . The focal subgroup theorem relates the ideas of transfer and fusion such as described i... | 2947865 | abstract_algebra |
Classification theorem in group theory
In mathematics, the Feit–Thompson theorem, or odd order theorem, states that every finite group of odd order is solvable. It was proved by Walter Feit and John Griggs Thompson (1962, 1963).
History.
The contrast that these results show between groups of odd and even order suggest... | 461822 | abstract_algebra |
In the area of modern algebra known as group theory, the Mathieu group "M22" is a sporadic simple group of order
27 · 32 · 5 · 7 · 11 = 443520
≈ 4×105.
History and properties.
"M22" is one of the 26 sporadic groups and was introduced by Mathieu (1861, 1873). It is a 3-fold transitive permutation group on 22 objects... | 1961349 | abstract_algebra |
Group of units of the ring of integers modulo n
In modular arithmetic, the integers coprime (relatively prime) to "n" from the set $\{0,1,\dots,n-1\}$ of "n" non-negative integers form a group under multiplication modulo "n", called the multiplicative group of integers modulo "n". Equivalently, the elements of this gro... | 367647 | abstract_algebra |
Subgroup mapped to itself under every automorphism of the parent group
In mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is mapped to itself by every automorphism of the parent group. Because every conjugation map is an inner automorphism, e... | 3273 | abstract_algebra |
In mathematics, specifically in ring theory, the simple modules over a ring "R" are the (left or right) modules over "R" that are non-zero and have no non-zero proper submodules. Equivalently, a module "M" is simple if and only if every cyclic submodule generated by a non-zero element of "M" equals "M". Simple modules ... | 14594 | abstract_algebra |
Mathematical group based upon a finite number of elements
In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations. Importa... | 145666 | abstract_algebra |
In group theory, a dicyclic group (notation Dic"n" or Q4"n", ⟨"n",2,2⟩) is a particular kind of non-abelian group of order 4"n" ("n" > 1). It is an extension of the cyclic group of order 2 by a cyclic group of order 2"n", giving the name "di-cyclic". In the notation of exact sequences of groups, this extension can b... | 89027 | abstract_algebra |
In mathematics, specifically in group theory, the Prüfer "p"-group or the p"-quasicyclic group or p"∞-group, Z("p"∞), for a prime number "p" is the unique "p"-group in which every element has "p" different "p"-th roots.
The Prüfer "p"-groups are countable abelian groups that are important in the classification of infi... | 964929 | abstract_algebra |
In mathematics, a Frobenius group is a transitive permutation group on a finite set, such that no non-trivial element
fixes more than one point and some non-trivial element fixes a point.
They are named after F. G. Frobenius.
Structure.
Suppose "G" is a Frobenius group consisting of permutations of a set "X". A subgro... | 473711 | abstract_algebra |
Automorphism group of the Klein quartic
In mathematics, the projective special linear group PSL(2, 7), isomorphic to GL(3, 2), is a finite simple group that has important applications in algebra, geometry, and number theory. It is the automorphism group of the Klein quartic as well as the symmetry group of the Fano pla... | 166097 | abstract_algebra |
In mathematics, or more specifically group theory, the omega and agemo subgroups described the so-called "power structure" of a finite "p"-group. They were introduced in where they were used to describe a class of finite "p"-groups whose structure was sufficiently similar to that of finite abelian "p"-groups, the so-ca... | 2672467 | abstract_algebra |
In mathematical group theory, the Thompson replacement theorem is a theorem about the existence of certain abelian subgroups of a "p"-group. The Glauberman replacement theorem is a generalization of it introduced by Glauberman (1968, Theorem 4.1).
Statement.
Suppose that "P" is a finite "p"-group for some prime "p", an... | 4677197 | abstract_algebra |
In mathematics, a class formation is a topological group acting on a module satisfying certain conditions. Class formations were introduced by Emil Artin and John Tate to organize the various Galois groups and modules that appear in class field theory.
Definitions.
A formation is a topological group "G" together with a... | 1104345 | abstract_algebra |
In mathematics, the O'Nan–Scott theorem is one of the most influential theorems of permutation group theory; the classification of finite simple groups is what makes it so useful. Originally the theorem was about maximal subgroups of the symmetric group. It appeared as an appendix to a paper by Leonard Scott written fo... | 4128190 | abstract_algebra |
Mathematics group theory concept
In mathematics, specifically group theory, the index of a subgroup "H" in a group "G" is the
number of left cosets of "H" in "G", or equivalently, the number of right cosets of "H" in "G".
The index is denoted $|G:H|$ or $[G:H]$ or $(G:H)$.
Because "G" is the disjoint union of the left... | 116973 | abstract_algebra |
In mathematics, the height of an element "g" of an abelian group "A" is an invariant that captures its divisibility properties: it is the largest natural number "N" such that the equation "Nx" = "g" has a solution "x" ∈ "A", or the symbol ∞ if there is no such "N". The "p"-height considers only divisibility properties ... | 3117425 | abstract_algebra |
In mathematics, Clifford theory, introduced by , describes the relation between representations of a group and those of a normal subgroup.
Alfred H. Clifford.
Alfred H. Clifford proved the following result on the restriction of finite-dimensional irreducible representations from a group "G" to a normal subgroup "N" of ... | 953329 | abstract_algebra |
In mathematical finite group theory, the concept of regular "p"-group captures some of the more important properties of abelian "p"-groups, but is general enough to include most "small" "p"-groups. Regular "p"-groups were introduced by Phillip Hall (1934).
Definition.
A finite "p"-group "G" is said to be regular if any... | 1984090 | abstract_algebra |
Algebraic structure
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic r... | 5508 | abstract_algebra |
Mathematical statement of uniqueness, except for an equivalent structure (equivalence relation)Two mathematical objects a and b are called "equal up to an equivalence relation R"
This figure of speech is mostly used in connection with expressions derived from equality, such as uniqueness or count.
For example, "x is un... | 22982 | abstract_algebra |
Five sporadic simple groups
In group theory, a topic in abstract algebra, the Mathieu groups are the five sporadic simple groups "M"11, "M"12, "M"22, "M"23 and "M"24 introduced by Mathieu (1861, 1873). They are multiply transitive permutation groups on 11, 12, 22, 23 or 24 objects. They are the first sporadic groups to... | 163223 | abstract_algebra |
Theorem in group theory
The Schur–Zassenhaus theorem is a theorem in group theory which states that if $G$ is a finite group, and $N$ is a normal subgroup whose order is coprime to the order of the quotient group $G/N$, then $G$ is a semidirect product (or split extension) of $N$ and $G/N$. An alternative statement of ... | 1319611 | abstract_algebra |
In group theory, a subfield of abstract algebra, a group cycle graph illustrates the various cycles of a group and is particularly useful in visualizing the structure of small finite groups.
A cycle is the set of powers of a given group element "a", where "an", the "n"-th power of an element "a" is defined as the prod... | 466450 | abstract_algebra |
3D symmetry group
A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation.
The group of all (not necessarily orientation preserving) symmetries is isomorphic to the group S4, the symmetric group of perm... | 689115 | abstract_algebra |
Concept in abstract algebra
In group theory, a branch of abstract algebra, extraspecial groups are analogues of the Heisenberg group over finite fields whose size is a prime. For each prime "p" and positive integer "n" there are exactly two (up to isomorphism) extraspecial groups of order "p"1+2"n". Extraspecial groups... | 1210564 | abstract_algebra |
Group of symmetries of an n-dimensional hypercube
In mathematics, a hyperoctahedral group is an important type of group that can be realized as the group of symmetries of a hypercube or of a cross-polytope. It was named by Alfred Young in 1930. Groups of this type are identified by a parameter n, the dimension of the h... | 2798556 | abstract_algebra |
Mathematical Tool in Group Theory
Named after the 19th century British mathematician Arthur Cayley, a Cayley table describes the structure of a finite group by arranging all the possible products of all the group's elements in a square table reminiscent of an addition or multiplication table. Many properties of a group... | 490452 | abstract_algebra |
In mathematics, the binary tetrahedral group, denoted 2T or ⟨2,3,3⟩, is a certain nonabelian group of order 24. It is an extension of the tetrahedral group T or (2,3,3) of order 12 by a cyclic group of order 2, and is the preimage of the tetrahedral group under the 2:1 covering homomorphism Spin(3) → SO(3) of the speci... | 1557508 | abstract_algebra |
In mathematics, and more specifically in homological algebra, the splitting lemma states that in any abelian category, the following statements are equivalent for a short exact sequence
$0 \longrightarrow A \mathrel{\overset{q}{\longrightarrow}} B \mathrel{\overset{r}{\longrightarrow}} C \longrightarrow 0.$
If any of... | 80160 | abstract_algebra |
Algebraic-group factorisation algorithms are algorithms for factoring an integer "N" by working in an algebraic group defined modulo "N" whose group structure is the direct sum of the 'reduced groups' obtained by performing the equations defining the group arithmetic modulo the unknown prime factors "p"1, "p"2, ... By ... | 1902874 | abstract_algebra |
Type of classification in algebra
In abstract algebra, a branch of mathematics, an Archimedean group is a linearly ordered group for which the Archimedean property holds: every two positive group elements are bounded by integer multiples of each other. The set R of real numbers together with the operation of addition a... | 128698 | abstract_algebra |
In mathematics, the binary octahedral group, name as 2O or ⟨2,3,4⟩ is a certain nonabelian group of order 48. It is an extension of the chiral octahedral group "O" or (2,3,4) of order 24 by a cyclic group of order 2, and is the preimage of the octahedral group under the 2:1 covering homomorphism $\operatorname{Spin}(3)... | 1557510 | abstract_algebra |
Mathematical abelian group
In mathematics, the Klein four-group is an abelian group with four elements, in which each element is self-inverse (composing it with itself produces the identity)
and in which composing any two of the three non-identity elements produces the third one.
It can be described as the symmetry gro... | 21489 | abstract_algebra |
Every subgroup of a cyclic group is cyclic, and if finite, its order divides its parent's
In abstract algebra, every subgroup of a cyclic group is cyclic. Moreover, for a finite cyclic group of order "n", every subgroup's order is a divisor of "n", and there is exactly one subgroup for each divisor. This result has bee... | 1541864 | abstract_algebra |
In mathematical finite group theory, the Thompson transitivity theorem gives conditions under which the centralizer of an abelian subgroup "A" acts transitively on certain subgroups normalized by "A". It originated in the proof of the odd order theorem by Feit and Thompson (1963), where it was used to prove the Thompso... | 3761379 | abstract_algebra |
In mathematics, the Walter theorem, proved by John H. Walter (1967, 1969), describes the finite groups whose Sylow 2-subgroup is abelian. used Bender's method to give a simpler proof.
Statement.
Walter's theorem states that if "G" is a finite group whose 2-sylow subgroups are abelian, then "G"/"O"("G") has a normal sub... | 3256934 | abstract_algebra |
Mathematic group theory
In mathematics, Burnside's theorem in group theory states that if "G" is a finite group of order $p^a q^b$ where "p" and "q" are prime numbers, and "a" and "b" are non-negative integers, then "G" is solvable. Hence each
non-Abelian finite simple group has order divisible by at least three distin... | 512355 | abstract_algebra |
Group that has an upper central series terminating with G
In mathematics, specifically group theory, a nilpotent group "G" is a group that has an upper central series that terminates with "G". Equivalently, its central series is of finite length or its lower central series terminates with {1}.
Intuitively, a nilpotent ... | 84093 | abstract_algebra |
In mathematics, specifically in group theory, the direct product is an operation that takes two groups "G" and "H" and constructs a new group, usually denoted "G" × "H". This operation is the group-theoretic analogue of the Cartesian product of sets and is one of several important notions of direct product in mathemati... | 812041 | abstract_algebra |
Subgroup of an abelian group consisting of all elements of finite order
In the theory of abelian groups, the torsion subgroup "AT" of an abelian group "A" is the subgroup of "A" consisting of all elements that have finite order (the torsion elements of "A"). An abelian group "A" is called a torsion group (or periodic g... | 83853 | abstract_algebra |
In mathematics, a locally compact group is a topological group "G" for which the underlying topology is locally compact and Hausdorff. Locally compact groups are important because many examples of groups that arise throughout mathematics are locally compact and such groups have a natural measure called the Haar measure... | 460810 | abstract_algebra |
A group is a set together with an associative operation which admits an identity element and such that every element has an inverse.
Throughout the article, we use $e$ to denote the identity element of a group.
* !$@ * See also* References
Basic definitions.
Subgroup. A subset $H$ of a group $(G, *)$ which remains a ... | 123200 | abstract_algebra |
In mathematics, the quasi-dihedral groups, also called semi-dihedral groups, are certain non-abelian groups of order a power of 2. For every positive integer "n" greater than or equal to 4, there are exactly four isomorphism classes of non-abelian groups of order 2"n" which have a cyclic subgroup of index 2. Two are we... | 333757 | abstract_algebra |
In mathematical group theory, Frobenius's theorem states that if "n" divides the order of a finite group "G", then the number of solutions of "x""n" = 1 is a multiple of "n". It was introduced by Frobenius (1903).
Statement.
A more general version of Frobenius's theorem states that if "C" is a conjugacy class with "h" ... | 4802593 | abstract_algebra |
Subset of a group that forms a group itself
In group theory, a branch of mathematics, given a group "G" under a binary operation ∗, a subset "H" of "G" is called a subgroup of "G" if "H" also forms a group under the operation ∗. More precisely, "H" is a subgroup of "G" if the restriction of ∗ to "H" × "H" is a group op... | 13862 | abstract_algebra |
In group theory, a field of mathematics, a double coset is a collection of group elements which are equivalent under the symmetries coming from two subgroups. More precisely, let "G" be a group, and let "H" and "K" be subgroups. Let "H" act on "G" by left multiplication and let "K" act on "G" by right multiplication. F... | 317574 | abstract_algebra |
Theorem that every subgroup of a free group is itself free
In group theory, a branch of mathematics, the Nielsen–Schreier theorem states that every subgroup of a free group is itself free. It is named after Jakob Nielsen and Otto Schreier.
Statement of the theorem.
A free group may be defined from a group presentation ... | 2228143 | abstract_algebra |
Disjoint, equal-size subsets of a group's underlying set
In mathematics, specifically group theory, a subgroup H of a group G may be used to decompose the underlying set of G into disjoint, equal-size subsets called cosets. There are "left cosets" and "right cosets". Cosets (both left and right) have the same number of... | 47025 | abstract_algebra |
Commutative group (mathematics)
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the ... | 1266 | abstract_algebra |
Factorization algorithm
In number theory, the general number field sieve (GNFS) is the most efficient classical algorithm known for factoring integers larger than 10100. Heuristically, its complexity for factoring an integer n (consisting of bits) is of the form
$\exp\left(\left((64/9)^{1/3}+o(1)\right)\left(\log n\rig... | 88079 | abstract_algebra |
Mathematical group
In mathematics, the outer automorphism group of a group, G, is the quotient, Aut("G") / Inn("G"), where Aut("G") is the automorphism group of G and Inn("G") is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted Out("G"). If Out("G") is trivial and G has a ... | 237359 | abstract_algebra |
Nonabelian group of order 120
In mathematics, the binary icosahedral group 2"I" or ⟨2,3,5⟩ is a certain nonabelian group of order 120.
It is an extension of the icosahedral group "I" or (2,3,5) of order 60 by the cyclic group of order 2, and is the preimage of the icosahedral group under the 2:1 covering homomorphism
$... | 736269 | abstract_algebra |
In mathematics, the Burnside ring of a finite group is an algebraic construction that encodes the different ways the group can act on finite sets. The ideas were introduced by William Burnside at the end of the nineteenth century. The algebraic ring structure is a more recent development, due to Solomon (1967).
Formal ... | 970531 | abstract_algebra |
In the area of modern algebra known as group theory, the Mathieu group "M12" is a sporadic simple group of order
12 · 11 · 10 · 9 · 8 = 26 · 33 · 5 · 11 = 95040.
History and properties.
"M12" is one of the 26 sporadic groups and was introduced by Mathieu (1861, 1873). It is a sharply 5-transitive permutation group o... | 1961348 | abstract_algebra |
In mathematics, especially in the area of algebra known as group theory, the holomorph of a group is a group that simultaneously contains (copies of) the group and its automorphism group. The holomorph provides interesting examples of groups, and allows one to treat group elements and group automorphisms in a uniform c... | 678049 | abstract_algebra |
In computational number theory, the index calculus algorithm is a probabilistic algorithm for computing discrete logarithms.
Dedicated to the discrete logarithm in $(\mathbb{Z}/q\mathbb{Z})^*$ where $q$ is a prime, index calculus leads to a family of algorithms adapted to finite fields and to some families of elliptic ... | 439747 | abstract_algebra |
In finite group theory, Jordan's theorem states that if a primitive permutation group "G" is a subgroup of the symmetric group "S""n" and contains a "p"-cycle for some prime number "p" < "n" − 2, then "G" is either the whole symmetric group "S""n" or the alternating group "A""n". It was first proved by Camille Jorda... | 3508255 | abstract_algebra |
In mathematics, in the realm of group theory, a class automorphism is an automorphism of a group that sends each element to within its conjugacy class. The class automorphisms form a subgroup of the automorphism group. Some facts:
For infinite groups, an example of a class automorphism that is not inner is the followin... | 1051878 | abstract_algebra |
Operation in group theory
In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product:
As with direct products, there is a natural equivalence between inner and outer semidirect products, and bot... | 24540 | abstract_algebra |
Mathematical connection between field theory and group theory
In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to group theory,... | 30755 | abstract_algebra |
Alternative mathematical ordering
In mathematics, a cyclic order is a way to arrange a set of objects in a circle.[nb] Unlike most structures in order theory, a cyclic order is not modeled as a binary relation, such as ""a" < "b"". One does not say that east is "more clockwise" than west. Instead, a cyclic order is ... | 180909 | abstract_algebra |
In mathematics, and in particular in combinatorics, the combinatorial number system of degree "k" (for some positive integer "k"), also referred to as combinadics, or the Macaulay representation of an integer, is a correspondence between natural numbers (taken to include 0) "N" and "k"-combinations. The combinations ar... | 348399 | abstract_algebra |
Method of graph decomposition
In graph theory, a bramble for an undirected graph G is a family of connected subgraphs of G that all touch each other: for every pair of disjoint subgraphs, there must exist an edge in G that has one endpoint in each subgraph. The "order" of a bramble is the smallest size of a hitting set... | 3829452 | abstract_algebra |
Non-abelian group of order eight
In group theory, the quaternion group Q8 (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to the eight-element subset
$\{1,i,j,k,-1,-i,-j,-k\}$ of the quaternions under multiplication. It is given by the group presentation
$\mathrm{Q}_8 = \langle \bar{e},i... | 88260 | abstract_algebra |
In mathematics, specifically group theory, a Hall subgroup of a finite group "G" is a subgroup whose order is coprime to its index. They were introduced by the group theorist Philip Hall (1928).
Definitions.
A Hall divisor (also called a unitary divisor) of an integer "n" is a divisor "d" of "n" such that
"d" and "n"/"... | 477233 | abstract_algebra |
In mathematics, specifically group theory, the free product is an operation that takes two groups "G" and "H" and constructs a new group "G" ∗ "H". The result contains both "G" and "H" as subgroups, is generated by the elements of these subgroups, and is the “universal” group having these properties, in the sense that ... | 157431 | abstract_algebra |
Studies linear representations of finite groups over a field K of positive characteristic p
Modular representation theory is a branch of mathematics, and is the part of representation theory that studies linear representations of finite groups over a field "K" of positive characteristic "p", necessarily a prime number.... | 347813 | abstract_algebra |
Problem a computer might be able to solve
In theoretical computer science, a computational problem is a problem that may be solved by an algorithm. For example, the problem of factoring
"Given a positive integer "n", find a nontrivial prime factor of "n"."
is a computational problem. A computational problem can be vie... | 930672 | abstract_algebra |
Algorithm for integer factorization
The Lenstra elliptic-curve factorization or the elliptic-curve factorization method (ECM) is a fast, sub-exponential running time, algorithm for integer factorization, which employs elliptic curves. For general-purpose factoring, ECM is the third-fastest known factoring method. The s... | 88699 | abstract_algebra |
Unsolved problem in mathematics:
Is every finite group the Galois group of a Galois extension of the rational numbers?
In Galois theory, the inverse Galois problem concerns whether or not every finite group appears as the Galois group of some Galois extension of the rational numbers $\mathbb{Q}$. This problem, first p... | 263714 | abstract_algebra |
Block design in combinatorial mathematics
In combinatorial mathematics, a Steiner system (named after Jakob Steiner) is a type of block design, specifically a with λ = 1 and "t" = 2 or (recently) "t" ≥ 2.
A Steiner system with parameters "t", "k", "n", written S("t","k","n"), is an "n"-element set "S" together with a s... | 13865 | abstract_algebra |
In graph theory, the degree diameter problem is the problem of finding the largest possible graph for a given maximum degree and diameter. The Moore bound sets limits on this, but for many years mathematicians in the field have been interested in a more precise answer. The table below gives current progress on this pro... | 2362270 | abstract_algebra |
In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a module, into simple pieces. The need for considering composition series in the context of modules arises from the fact that many naturally occurring modules are not semisimple, hence cannot be decomposed in... | 136992 | abstract_algebra |
In the mathematical discipline of graph theory, a 3-dimensional matching is a generalization of bipartite matching (also known as 2-dimensional matching) to 3-partite hypergraphs, which consist of hyperedges each of which contains 3 vertices (instead of edges containing 2 vertices in a usual graph).
3-dimensional match... | 2598868 | abstract_algebra |
The problem of inverting exponentiation in finite groups
In mathematics, for given real numbers "a" and "b", the logarithm log"b" "a" is a number "x" such that "b""x" = "a". Analogously, in any group "G", powers "b""k" can be defined for all integers "k", and the discrete logarithm log"b" "a" is an integer "k" such tha... | 99905 | abstract_algebra |
Permutation group that preserves no non-trivial partition
In mathematics, a permutation group "G" acting on a non-empty finite set "X" is called primitive if "G" acts transitively on "X" and the only partitions the "G"-action preserves are the trivial partitions into either a single set or into |"X"| singleton sets. Ot... | 662598 | abstract_algebra |
In mathematics, the ping-pong lemma, or table-tennis lemma, is any of several mathematical statements that ensure that several elements in a group acting on a set freely generates a free subgroup of that group.
History.
The ping-pong argument goes back to the late 19th century and is commonly attributed to Felix Klein ... | 2274857 | abstract_algebra |
Fundamental result in the branch of mathematics known as character theory
Brauer's theorem on induced characters, often known as Brauer's induction theorem, and named after Richard Brauer, is a basic result in the branch of mathematics known as character theory, within representation theory of a finite group.
Backgroun... | 713702 | abstract_algebra |
Non-commutative group with 6 elements
In mathematics, D3 (sometimes alternatively denoted by D6) is the dihedral group of degree 3 and order 6. It equals the symmetric group S3. It is also the smallest non-abelian group.
This page illustrates many group concepts using this group as example.
Symmetry groups.
The dihedra... | 599449 | abstract_algebra |
The following list in mathematics contains the finite groups of small order up to group isomorphism.
Counts.
For "n" = 1, 2, … the number of nonisomorphic groups of order "n" is
1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14, 1, 5, 1, 5, ... (sequence in the OEIS)
For labeled groups, see OEIS: .
Glossary.
Each group ... | 24355 | abstract_algebra |
Zero divisors in a module
In mathematics, specifically in ring theory, a torsion element is an element of a module that yields zero when multiplied by some non-zero-divisor of the ring. The torsion submodule of a module is the submodule formed by the torsion elements. A torsion module is a module that equals its torsio... | 711423 | abstract_algebra |
Mathematical group with trivial abelianization
In mathematics, more specifically in group theory, a group is said to be perfect if it equals its own commutator subgroup, or equivalently, if the group has no non-trivial abelian quotients (equivalently, its abelianization, which is the universal abelian quotient, is triv... | 224545 | abstract_algebra |
Math theorem in the field of representation theory
Itô's theorem is a result in the mathematical discipline of representation theory due to Noboru Itô. It generalizes the well-known result that the dimension of an irreducible representation of a group must divide the order of that group.
Statement.
Given an irreducible... | 2404482 | abstract_algebra |
In mathematics and in particular in combinatorics, the Lehmer code is a particular way to encode each possible permutation of a sequence of "n" numbers. It is an instance of a scheme for numbering permutations and is an example of an inversion table.
The Lehmer code is named in reference to Derrick Henry Lehmer, but th... | 2958741 | abstract_algebra |
Mathematical theorem
In mathematics, the Krull–Schmidt theorem states that a group subjected to certain finiteness conditions on chains of subgroups, can be uniquely written as a finite direct product of indecomposable subgroups.
Definitions.
We say that a group "G" satisfies the ascending chain condition (ACC) on subg... | 2483140 | abstract_algebra |
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