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c_uu8kkbrbucxs | In mathematics, in the field of group theory, a locally finite group is a type of group that can be studied in ways analogous to a finite group. Sylow subgroups, Carter subgroups, and abelian subgroups of locally finite groups have been studied. The concept is credited to work in the 1930s by Russian mathematician Serg... | Locally finite group |
c_u3azkpva40wg | In mathematics, in the field of group theory, a metanilpotent group is a group that is nilpotent by nilpotent. In other words, it has a normal nilpotent subgroup such that the quotient group is also nilpotent. In symbols, G {\displaystyle G} is metanilpotent if there is a normal subgroup N {\displaystyle N} such that b... | Metanilpotent group |
c_tbfx8tlzrmvn | In mathematics, in the field of group theory, a modular subgroup is a subgroup that is a modular element in the lattice of subgroups, where the meet operation is defined by the intersection and the join operation is defined by the subgroup generated by the union of subgroups. By the modular property of groups, every qu... | Modular subgroup |
c_5objshrsw15r | In mathematics, in the field of group theory, a paranormal subgroup is a subgroup such that the subgroup generated by it and any conjugate of it, is also generated by it and a conjugate of it within that subgroup. In symbols, H {\displaystyle H} is paranormal in G {\displaystyle G} if given any g {\displaystyle g} in G... | Paranormal subgroup |
c_i0ezgqosx1sg | 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 c... | PT-group |
c_za5fl0mm9ywp | That is, H {\displaystyle H} and K {\displaystyle K} as subgroups of G {\displaystyle G} are said to commute if HK = KH, that is, any element of the form h k {\displaystyle hk} with h ∈ H {\displaystyle h\in H} and k ∈ K {\displaystyle k\in K} can be written in the form k ′ h ′ {\displaystyle k'h'} where k ′ ∈ K {\disp... | PT-group |
c_we4zhwnl60g5 | For instance, any extension of a cyclic p {\displaystyle p} -group by another cyclic p {\displaystyle p} -group for the same (odd) prime has the property that all its subgroups are quasinormal. However, not all of its subgroups need be normal. Every quasinormal subgroup is a modular subgroup, that is, a modular element... | PT-group |
c_81ro0fwtag6f | This follows from the modular property of groups. If all subgroups are quasinormal, then the group is called an Iwasawa group—sometimes also called a modular group, although this latter term has other meanings. In any group, every quasinormal subgroup is ascendant. A conjugate permutable subgroup is one that commutes w... | PT-group |
c_6ku8nghlnhjy | In mathematics, in the field of group theory, a subgroup H of a given group G is a subnormal subgroup of G if there is a finite chain of subgroups of the group, each one normal in the next, beginning at H and ending at G. In notation, H {\displaystyle H} is k {\displaystyle k} -subnormal in G {\displaystyle G} if there... | Subnormal subgroup |
c_e7vjp9seib5r | Every quasinormal subgroup, and, more generally, every conjugate-permutable subgroup, of a finite group is subnormal. Every pronormal subgroup that is also subnormal, is normal. In particular, a Sylow subgroup is subnormal if and only if it is normal. | Subnormal subgroup |
c_fc0s5oveuno1 | Every 2-subnormal subgroup is a conjugate-permutable subgroup.The property of subnormality is transitive, that is, a subnormal subgroup of a subnormal subgroup is subnormal. The relation of subnormality can be defined as the transitive closure of the relation of normality. If every subnormal subgroup of G is normal in ... | Subnormal subgroup |
c_3jjqi01h7suu | In mathematics, in the field of group theory, a subgroup H {\displaystyle H} of a group G {\displaystyle G} is called c-normal if there is a normal subgroup T {\displaystyle T} of G {\displaystyle G} such that H T = G {\displaystyle HT=G} and the intersection of H {\displaystyle H} and T {\displaystyle T} lies inside t... | C-normal subgroup |
c_ot6jct6jamm2 | In mathematics, in the field of group theory, a subgroup H {\displaystyle H} of a group G {\displaystyle G} is said to be weakly normal if whenever H g ≤ N G ( H ) {\displaystyle H^{g}\leq N_{G}(H)} , we have g ∈ N G ( H ) {\displaystyle g\in N_{G}(H)} . Every pronormal subgroup is weakly normal. | Weakly normal subgroup |
c_higmu8dgv5z7 | In mathematics, in the field of group theory, a subgroup H {\displaystyle H} of a group G {\displaystyle G} is termed malnormal if for any x {\displaystyle x} in G {\displaystyle G} but not in H {\displaystyle H} , H {\displaystyle H} and x H x − 1 {\displaystyle xHx^{-1}} intersect in the identity element.Some facts a... | Malnormal subgroup |
c_3xqpaq7vb1mw | Every malnormal subgroup is a special type of C-group called a trivial intersection subgroup or TI subgroup.When G is finite, a malnormal subgroup H distinct from 1 and G is called a "Frobenius complement". The set N of elements of G which are, either equal to 1, or non-conjugate to any element of H, is a normal subgro... | Malnormal subgroup |
c_77kfro3wh91s | In mathematics, in the field of group theory, a subgroup of a group is said to be ascendant if there is an ascending series starting from the subgroup and ending at the group, such that every term in the series is a normal subgroup of its successor. The series may be infinite. If the series is finite, then the subgroup... | Ascendant subgroup |
c_gs42fc7taoku | Here are some properties of ascendant subgroups: Every subnormal subgroup is ascendant; every ascendant subgroup is serial. In a finite group, the properties of being ascendant and subnormal are equivalent. An arbitrary intersection of ascendant subgroups is ascendant. Given any subgroup, there is a minimal ascendant s... | Ascendant subgroup |
c_5svuqphj8x45 | In mathematics, in the field of group theory, a subgroup of a group is said to be conjugacy-closed if any two elements of the subgroup that are conjugate in the group are also conjugate in the subgroup. An alternative characterization of conjugacy-closed normal subgroups is that all class automorphisms of the whole gro... | Conjugacy-closed subgroup |
c_mlhcjukhmebx | The property of being conjugacy-closed is transitive, that is, every conjugacy-closed subgroup of a conjugacy-closed subgroup is conjugacy-closed.The property of being conjugacy-closed is sometimes also termed as being conjugacy stable. It is a known result that for finite field extensions, the general linear group of ... | Conjugacy-closed subgroup |
c_oqcjixy5rbbk | In mathematics, in the field of group theory, a subgroup of a group is said to be descendant if there is a descending series starting from the subgroup and ending at the group, such that every term in the series is a normal subgroup of its predecessor. The series may be infinite. If the series is finite, then the subgr... | Descendant subgroup |
c_mu92mmf806kg | In mathematics, in the field of group theory, a subgroup of a group is said to be fully normalized if every automorphism of the subgroup lifts to an inner automorphism of the whole group. Another way of putting this is that the natural embedding from the Weyl group of the subgroup to its automorphism group is surjectiv... | Fully normalized subgroup |
c_dhjlqa0k4yog | A natural construction for this is the holomorph, which is its semidirect product with its automorphism group. A complete group is fully normalized in any bigger group in which it is embedded because every automorphism of it is inner. Every fully normalized subgroup has the automorphism extension property. | Fully normalized subgroup |
c_6223phdc8ytb | In mathematics, in the field of group theory, a subgroup of a group is said to be polynormal if its closure under conjugation by any element of the group can also be achieved via closure by conjugation by some element in the subgroup generated. In symbols, a subgroup H {\displaystyle H} of a group G {\displaystyle G} i... | Polynormal subgroup |
c_eocmrzicjdg4 | In mathematics, in the field of group theory, a subgroup of a group is said to be transitively normal in the group if every normal subgroup of the subgroup is also normal in the whole group. In symbols, H {\displaystyle H} is a transitively normal subgroup of G {\displaystyle G} if for every K {\displaystyle K} normal ... | Transitively normal subgroup |
c_n4espps63r3s | Thus, every central subgroup is transitively normal. A transitively normal subgroup of a transitively normal subgroup is transitively normal. A transitively normal subgroup is normal. | Transitively normal subgroup |
c_c6gamgr22fjt | In mathematics, in the field of group theory, a subgroup of a group is said to have the Congruence Extension Property or to be a CEP subgroup if every congruence on the subgroup lifts to a congruence of the whole group. Equivalently, every normal subgroup of the subgroup arises as the intersection with the subgroup of ... | CEP subgroup |
c_0f9b3xhjf0g0 | In mathematics, in the field of group theory, a subgroup of a group is termed a retract if there is an endomorphism of the group that maps surjectively to the subgroup and is identity on the subgroup. In symbols, H {\displaystyle H} is a retract of G {\displaystyle G} if and only if there is an endomorphism σ: G → G {\... | Retract (group theory) |
c_7ftm0yppyaum | Every direct factor is a retract. Conversely, any retract which is a normal subgroup is a direct factor. Every retract has the congruence extension property. Every regular factor, and in particular, every free factor, is a retract. | Retract (group theory) |
c_1q8mkazd6vf5 | In mathematics, in the field of group theory, a subgroup of a group is termed central if it lies inside the center of the group. Given a group G {\displaystyle G} , the center of G {\displaystyle G} , denoted as Z ( G ) {\displaystyle Z(G)} , is defined as the set of those elements of the group which commute with every... | Central subgroup |
c_04o6k6jrps79 | Central subgroups have the following properties: They are abelian groups (because, in particular, all elements of the center must commute with each other). They are normal subgroups. They are central factors, and are hence transitively normal subgroups. | Central subgroup |
c_roqd9ojpv2oh | In mathematics, in the field of group theory, an FC-group is a group in which every conjugacy class of elements has finite cardinality. The following are some facts about FC-groups: Every finite group is an FC-group. Every abelian group is an FC-group. The following property is stronger than the property of being FC: e... | FC-group |
c_rttw3h2yw235 | In mathematics, in the field of group theory, especially in the study of p-groups and pro-p-groups, the concept of powerful p-groups plays an important role. They were introduced in (Lubotzky & Mann 1987), where a number of applications are given, including results on Schur multipliers. Powerful p-groups are used in th... | Powerful p-group |
c_gofed38afoq8 | In mathematics, in the field of group theory, the norm of a group is the intersection of the normalizers of all its subgroups. This is also termed the Baer norm, after Reinhold Baer. The following facts are true for the Baer norm: It is a characteristic subgroup. | Norm (group) |
c_x8ar2inaznhe | It contains the center of the group. It is contained inside the second term of the upper central series. It is a Dedekind group, so is either abelian or has a direct factor isomorphic to the quaternion group. If it contains an element of infinite order, then it is equal to the center of the group. | Norm (group) |
c_6v1qhdbpvvcb | In mathematics, in the field of group theory, the perfect core (or perfect radical) of a group is its largest perfect subgroup. Its existence is guaranteed by the fact that the subgroup generated by a family of perfect subgroups is again a perfect subgroup. The perfect core is also the point where the transfinite deriv... | Perfect core |
c_r6czvsfo6pv9 | Every solvable group is hypoabelian, and so is every free group. More generally, every residually solvable group is hypoabelian. The quotient of a group G by its perfect core is hypoabelian, and is called the hypoabelianization of G. == References == | Perfect core |
c_5xbix8c3l8cq | In mathematics, in the field of harmonic analysis, an oscillatory integral operator is an integral operator of the form T λ u ( x ) = ∫ R n e i λ S ( x , y ) a ( x , y ) u ( y ) d y , x ∈ R m , y ∈ R n , {\displaystyle T_{\lambda }u(x)=\int _{\mathbb {R} ^{n}}e^{i\lambda S(x,y)}a(x,y)u(y)\,dy,\qquad x\in \mathbb {R} ^{... | Oscillatory integral operator |
c_arpwi6wxlf5h | In mathematics, in the field of harmonic analysis, the van der Corput lemma is an estimate for oscillatory integrals named after the Dutch mathematician J. G. van der Corput. The following result is stated by E. Stein:Suppose that a real-valued function ϕ ( x ) {\displaystyle \phi (x)} is smooth in an open interval ( a... | Van der Corput lemma (harmonic analysis) |
c_a9qpwbrtjup6 | In mathematics, in the field of homological algebra, given an abelian category C {\displaystyle {\mathcal {C}}} having enough injectives and an additive (covariant) functor F: C → D {\displaystyle F:{\mathcal {C}}\to {\mathcal {D}}} ,an acyclic object with respect to F {\displaystyle F} , or simply an F {\displaystyle ... | Acyclic object |
c_9vog88bb0s3w | In mathematics, in the field of homological algebra, the Grothendieck spectral sequence, introduced by Alexander Grothendieck in his Tôhoku paper, is a spectral sequence that computes the derived functors of the composition of two functors G ∘ F {\displaystyle G\circ F} , from knowledge of the derived functors of F {\d... | Grothendieck spectral sequence |
c_wq7ju48a35ki | In mathematics, in the field of measure theory, τ-additivity is a certain property of measures on topological spaces. A measure or set function μ {\displaystyle \mu } on a space X {\displaystyle X} whose domain is a sigma-algebra Σ {\displaystyle \Sigma } is said to be τ-additive if for any upward-directed family G ⊆ Σ... | Τ-additivity |
c_9xlg4pyvcicm | In mathematics, in the field of number theory, the Ramanujan–Nagell equation is an equation between a square number and a number that is seven less than a power of two. It is an example of an exponential Diophantine equation, an equation to be solved in integers where one of the variables appears as an exponent. The eq... | Ramanujan–Nagell equation |
c_2nvo8v5lw9zx | In mathematics, in the field of ordinary differential equations, Sturm separation theorem, named after Jacques Charles François Sturm, describes the location of roots of solutions of homogeneous second order linear differential equations. Basically the theorem states that given two linear independent solutions of such ... | Sturm separation theorem |
c_ei96b6zs9ij8 | In mathematics, in the field of ordinary differential equations, a nontrivial solution to an ordinary differential equation F ( x , y , y ′ , … , y ( n − 1 ) ) = y ( n ) x ∈ [ 0 , + ∞ ) {\displaystyle F(x,y,y',\ \dots ,\ y^{(n-1)})=y^{(n)}\quad x\in [0,+\infty )} is called oscillating if it has an infinite number of ro... | Oscillation theory |
c_gkokhha4l46e | In mathematics, in the field of ordinary differential equations, the Sturm–Picone comparison theorem, named after Jacques Charles François Sturm and Mauro Picone, is a classical theorem which provides criteria for the oscillation and non-oscillation of solutions of certain linear differential equations in the real doma... | Sturm-Picone comparison theorem |
c_htdlu48s46zk | In mathematics, in the field of p-adic analysis, the Volkenborn integral is a method of integration for p-adic functions. | Volkenborn integral |
c_d0oqjg0gbkz7 | In mathematics, in the field of potential theory, the fine topology is a natural topology for setting the study of subharmonic functions. In the earliest studies of subharmonic functions, namely those for which Δ u ≥ 0 , {\displaystyle \Delta u\geq 0,} where Δ {\displaystyle \Delta } is the Laplacian, only smooth funct... | Classical fine topology |
c_licqnlw1siai | In mathematics, in the field of ring theory, a lattice is a module over a ring which is embedded in a vector space over a field, giving an algebraic generalisation of the way a lattice group is embedded in a real vector space. | Lattice (module) |
c_y99cum4u6ly7 | In mathematics, in the field of topology, a topological space X {\displaystyle X} is said to be collectionwise Hausdorff if given any closed discrete subset of X {\displaystyle X} , there is a pairwise disjoint family of open sets with each point of the discrete subset contained in exactly one of the open sets.Here a s... | Collectionwise Hausdorff space |
c_9l3jlf0bsndy | In mathematics, in the field of topology, a topological space is called supercompact if there is a subbasis such that every open cover of the topological space from elements of the subbasis has a subcover with at most two subbasis elements. Supercompactness and the related notion of superextension was introduced by J. ... | Supercompact space |
c_y196jgjoqtpy | In mathematics, in the field of topology, a topological space is said to be a Volterra space if any finite intersection of dense Gδ subsets is dense. Every Baire space is Volterra, but the converse is not true. In fact, any metrizable Volterra space is Baire. The name refers to a paper of Vito Volterra in which he uses... | Volterra space |
c_idughddes9ys | In mathematics, in the field of topology, a topological space is said to be a shrinking space if every open cover admits a shrinking. A shrinking of an open cover is another open cover indexed by the same indexing set, with the property that the closure of each open set in the shrinking lies inside the corresponding or... | Shrinking space |
c_yr8zp9fo3zfd | In mathematics, in the field of topology, a topological space is said to be a-paracompact if every open cover of the space has a locally finite refinement. In contrast to the definition of paracompactness, the refinement is not required to be open. Every paracompact space is a-paracompact, and in regular spaces the two... | A-paracompact space |
c_6l38iz737z1n | In mathematics, in the field of topology, a topological space is said to be hemicompact if it has a sequence of compact subsets such that every compact subset of the space lies inside some compact set in the sequence. Clearly, this forces the union of the sequence to be the whole space, because every point is compact a... | Hemicompact space |
c_8en2839ol5tw | In mathematics, in the field of topology, a topological space is said to be locally Hausdorff if every point has a neighbourhood that is a Hausdorff space under the subspace topology. | Locally Hausdorff space |
c_tzd31a3ffhtf | In mathematics, in the field of topology, a topological space is said to be pseudocompact if its image under any continuous function to R is bounded. Many authors include the requirement that the space be completely regular in the definition of pseudocompactness. Pseudocompact spaces were defined by Edwin Hewitt in 194... | Pseudocompact space |
c_u91ijw2q2k1f | In mathematics, in the field of topology, a topological space is said to be pseudonormal if given two disjoint closed sets in it, one of which is countable, there are disjoint open sets containing them. Note the following: Every normal space is pseudonormal. Every pseudonormal space is regular.An example of a pseudonor... | Pseudonormal space |
c_09k2uyv37rnq | In mathematics, in the field of topology, a topological space is said to be realcompact if it is completely regular Hausdorff and it contains every point of its Stone–Čech compactification which is real (meaning that the quotient field at that point of the ring of real functions is the reals). Realcompact spaces have a... | Realcompact space |
c_n8mpl9komvp7 | In mathematics, in the field of tropical analysis, the log semiring is the semiring structure on the logarithmic scale, obtained by considering the extended real numbers as logarithms. That is, the operations of addition and multiplication are defined by conjugation: exponentiate the real numbers, obtaining a positive ... | Log semiring |
c_e23sensbiz0c | If not qualified, the base is conventionally taken to be e or 1/e, which corresponds to e with a negative. The log semiring has the tropical semiring as limit ("tropicalization", "dequantization") as the base goes to infinity b → ∞ {\displaystyle b\to \infty } (max-plus semiring) or to zero b → 0 {\displaystyle b\to 0}... | Log semiring |
c_4pl9o79qvj3e | Notably, the addition operation, logadd (for multiple terms, LogSumExp) can be viewed as a deformation of maximum or minimum. The log semiring has applications in mathematical optimization, since it replaces the non-smooth maximum and minimum by a smooth operation. The log semiring also arises when working with numbers... | Log semiring |
c_ejr9af77c0b0 | In mathematics, in the fields of multilinear algebra and representation theory, the principal invariants of the second rank tensor A {\displaystyle \mathbf {A} } are the coefficients of the characteristic polynomial p ( λ ) = det ( A − λ I ) {\displaystyle \ p(\lambda )=\det(\mathbf {A} -\lambda \mathbf {I} )} ,where I... | Invariants of tensors |
c_nue3tox0x2wx | In mathematics, in the framework of one-universe foundation for category theory, the term "conglomerate" is applied to arbitrary sets as a contraposition to the distinguished sets that are elements of a Grothendieck universe. | Conglomerate (mathematics) |
c_rev3g6b2zcih | In mathematics, in the phase portrait of a dynamical system, a heteroclinic orbit (sometimes called a heteroclinic connection) is a path in phase space which joins two different equilibrium points. If the equilibrium points at the start and end of the orbit are the same, the orbit is a homoclinic orbit. Consider the co... | Heteroclinic orbit |
c_winrzq0wu235 | In mathematics, in the realm of abelian group theory, a group is said to be algebraically compact if it is a direct summand of every abelian group containing it as a pure subgroup. Equivalent characterizations of algebraic compactness: The reduced part of the group is Hausdorff and complete in the Z {\displaystyle \mat... | Algebraically compact group |
c_60r84szivza3 | In mathematics, in the realm of abstract algebra, a radical polynomial is a multivariate polynomial over a field that can be expressed as a polynomial in the sum of squares of the variables. That is, if k {\displaystyle k} is a polynomial ring, the ring of radical polynomials is the subring generated by the polynomial... | Radical polynomial |
c_cnjnka8y0out | The ring of radical polynomials is a graded subalgebra of the ring of all polynomials. The standard separation of variables theorem asserts that every polynomial can be expressed as a finite sum of terms, each term being a product of a radical polynomial and a harmonic polynomial. This is equivalent to the statement th... | Radical polynomial |
c_44d5of3gz177 | 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: Every inner automorphism is a class automorphism. Every class automorphism is a family au... | Class automorphism |
c_dnwuo6ocprja | Under a quotient map, class automorphisms go to class automorphisms. Every class automorphism is an IA automorphism, that is, it acts as identity on the abelianization. Every class automorphism is a center-fixing automorphism, that is, it fixes all points in the center. | Class automorphism |
c_ncjquguih038 | Normal subgroups are characterized as subgroups invariant under class automorphisms.For infinite groups, an example of a class automorphism that is not inner is the following: take the finitary symmetric group on countably many elements and consider conjugation by an infinitary permutation. This conjugation defines an ... | Class automorphism |
c_ndbxto6tp963 | This is essentially because the infinitary permutation takes permutations of finite supports to permutations of finite support. For finite groups, the classical example is a group of order 32 obtained as the semidirect product of the cyclic ring on 8 elements, by its group of units acting via multiplication. Finding a ... | Class automorphism |
c_rjakee74yvxr | In mathematics, in the realm of group theory, a countable group is said to be SQ-universal if every countable group can be embedded in one of its quotient groups. SQ-universality can be thought of as a measure of largeness or complexity of a group. | SQ-universal group |
c_o9vn1a7ehsib | In mathematics, in the realm of group theory, a group is said to be a CA-group or centralizer abelian group if the centralizer of any nonidentity element is an abelian subgroup. Finite CA-groups are of historical importance as an early example of the type of classifications that would be used in the Feit–Thompson theor... | CA-group |
c_q5ij89muubz1 | In mathematics, in the realm of group theory, a group is said to be capable if it occurs as the inner automorphism group of some group. These groups were first studied by Reinhold Baer, who showed that a finite abelian group is capable if and only if it is a product of cyclic groups of orders n1,...,nk where ni divides... | Capable group |
c_0g18cgoiu6rg | In mathematics, in the realm of group theory, a group is said to be critical if it is not in the variety generated by all its proper subquotients, which includes all its subgroups and all its quotients. Any finite monolithic A-group is critical. This result is due to Kovacs and Newman. The variety generated by a finite... | Critical group |
c_zf4eddj6blnd | In mathematics, in the realm of group theory, a group is said to be parafree if its quotients by the terms of its lower central series are the same as those of a free group and if it is residually nilpotent (the intersection of the terms of its lower central series is trivial). Parafree groups share many properties wit... | Parafree group |
c_9upekhjgxj0w | In mathematics, in the realm of group theory, a group is said to be superperfect when its first two homology groups are trivial: H1(G, Z) = H2(G, Z) = 0. This is stronger than a perfect group, which is one whose first homology group vanishes. In more classical terms, a superperfect group is one whose abelianization and... | Superperfect group |
c_4wigtbpbtm0c | In mathematics, in the realm of group theory, a group is said to be thin if there is a finite upper bound on the girth of the Cayley graph induced by any finite generating set. The group is called fat if it is not thin. Given any generating set of the group, we can consider a graph whose vertices are elements of the gr... | Thin group (combinatorial group theory) |
c_pur5ytsjodrq | Paths in the graph correspond to words in the generators. If the graph has a cycle of a given length, it has a cycle of the same length containing the identity element. Thus, the girth of the graph corresponds to the minimum length of a nontrivial word that reduces to the identity. | Thin group (combinatorial group theory) |
c_1hptv6mi9dz3 | A nontrivial word is a word that, if viewed as a word in the free group, does not reduce to the identity. If the graph has no cycles, its girth is set to be infinity. | Thin group (combinatorial group theory) |
c_fi0bw7pdk1a5 | The girth depends on the choice of generating set. A thin group is a group where the girth has an upper bound for all finite generating sets. | Thin group (combinatorial group theory) |
c_lshzfsa07lrk | Some facts about thin and fat groups and about girths: Every finite group is thin. Every free group is fat. The girth of a cyclic group equals its order. | Thin group (combinatorial group theory) |
c_rbpwkmg78mym | The girth of a noncyclic abelian group is at most 4, because any two elements commute and the commutation relation gives a nontrivial word. The girth of the dihedral group is 2. Every nilpotent group, and more generally, every solvable group, is thin. | Thin group (combinatorial group theory) |
c_go24mp21ysex | In mathematics, in the realm of group theory, a power automorphism of a group is an automorphism that takes each subgroup of the group to within itself. It is worth noting that the power automorphism of an infinite group may not restrict to an automorphism on each subgroup. For instance, the automorphism on rational nu... | Power automorphism |
c_e8zghmy9muek | This explains the choice of the term power. The power automorphisms of a group form a subgroup of the whole automorphism group. This subgroup is denoted as P o t ( G ) {\displaystyle Pot(G)} where G {\displaystyle G} is the group. | Power automorphism |
c_56isjzd7lp5e | A universal power automorphism is a power automorphism where the power to which each element is raised is the same. For instance, each element may go to its cube. Here are some facts about the powering index: The powering index must be relatively prime to the order of each element. | Power automorphism |
c_4r9x7zfjx2kg | In particular, it must be relatively prime to the order of the group, if the group is finite. If the group is abelian, any powering index works. If the powering index 2 or -1 works, then the group is abelian.The group of power automorphisms commutes with the group of inner automorphisms when viewed as subgroups of the ... | Power automorphism |
c_kvouftmyierl | In mathematics, in the realm of group theory, a quotientable automorphism of a group is an automorphism that takes every normal subgroup to within itself. As a result, it gives a corresponding automorphism for every quotient group. All family automorphisms are quotientable, and particularly, all class automorphisms and... | Quotientable automorphism |
c_7illoq0ka33y | In mathematics, in the realm of group theory, a subgroup of a group is said to be centrally closed if the centralizer of any nonidentity element of the subgroup lies inside the subgroup. Some facts about centrally closed subgroups: Every malnormal subgroup is centrally closed. Every Frobenius kernel is centrally closed... | Centrally closed subgroup |
c_9ck0shnh05o4 | In mathematics, in the realm of group theory, an IA automorphism of a group is an automorphism that acts as identity on the abelianization. The abelianization of a group is its quotient by its commutator subgroup. An IA automorphism is thus an automorphism that sends each coset of the commutator subgroup to itself. The... | IA automorphism |
c_ywslwz9laj2l | In mathematics, in the realm of group theory, the stability group of subnormal series is the group of automorphisms that act as identity on each quotient group. | Stability group |
c_zzs11oixr747 | In mathematics, in the realm of group theory, the term complemented group is used in two distinct, but similar ways. In (Hall 1937), a complemented group is one in which every subgroup has a group-theoretic complement. Such groups are called completely factorizable groups in the Russian literature, following (Baeva 195... | Complemented group |
c_hror08exg1i5 | Hall's definition required in addition that H and K permute, that is, that HK = { hk: h in H, k in K } form a subgroup. Such groups are also called K-groups in the Italian and lattice theoretic literature, such as (Schmidt 1994, pp. | Complemented group |
c_1s0d47t09vsa | 114–121, Chapter 3.1). The Frattini subgroup of a K-group is trivial; if a group has a core-free maximal subgroup that is a K-group, then it itself is a K-group; hence subgroups of K-groups need not be K-groups, but quotient groups and direct products of K-groups are K-groups, (Schmidt 1994, pp. 115–116). | Complemented group |
c_duckoza7ls1a | In (Costantini & Zacher 2004) it is shown that every finite simple group is a complemented group. Note that in the classification of finite simple groups, K-group is more used to mean a group whose proper subgroups only have composition factors amongst the known finite simple groups. An example of a group that is not c... | Complemented group |
c_8iijx4ecsmyz | In mathematics, in the realm of point-set topology, a Toronto space is a topological space that is homeomorphic to every proper subspace of the same cardinality. There are five homeomorphism classes of countable Toronto spaces, namely: the discrete topology, the indiscrete topology, the cofinite topology and the upper ... | Toronto space |
c_aszcifno0bps | In mathematics, in the realm of topology, a paranormal space (Nyikos 1984) is a topological space in which every countable discrete collection of closed sets has a locally finite open expansion. | Paranormal space |
c_58rtbvu1h75c | In mathematics, in the representation theory of algebraic groups, a Grosshans subgroup, named after Frank Grosshans, is an algebraic subgroup of an algebraic group that is an observable subgroup for which the ring of functions on the quotient variety is finitely generated. | Grosshans subgroup |
c_hlbznmw7p12p | In mathematics, in the representation theory of algebraic groups, a linear representation of an algebraic group is said to be rational if, viewed as a map from the group to the general linear group, it is a rational map of algebraic varieties. Finite direct sums and products of rational representations are rational. A ... | Rational representation |
c_2kv8tyyzifrg | In mathematics, in the representation theory of algebraic groups, an observable subgroup is an algebraic subgroup of a linear algebraic group whose every finite-dimensional rational representation arises as the restriction to the subgroup of a finite-dimensional rational representation of the whole group. An equivalent... | Observable subgroup |
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