# Search results

• ...a representation|degree]] of that representation, i.e., the ''dimension'' of the vector space on which the action is happening. For an irreducible repre ===Typical context: finite group and splitting field===
15 KB (2,038 words) - 02:28, 28 May 2013
• ...group]], as well as the relation among them. More specifically, given a [[finite group]]: ...gruence''': The number of [itex]p[/itex]-Sylow subgroups divides the index of any [itex]p[/itex]-Sylow subgroup and is also congruent to [itex]1[/itex] m
4 KB (602 words) - 05:42, 24 September 2016
• ...hrase|[[quick phrase::invariant under all automorphisms]], [[quick phrase::automorphism-invariant]], [[quick phrase::strongly normal]], [[quick phrase::normal unde Below are many '''equivalent''' definitions of characteristic subgroup.
40 KB (4,850 words) - 18:25, 30 November 2019
• {{automorphism property implication| stronger = finite-extensible automorphism|
3 KB (296 words) - 19:20, 30 May 2009
• Let [itex]P[/itex] be a finite [itex]p[/itex]-group, i.e., a [[group of prime power order]] where the underlying prime is [itex]p[/itex]. Suppose t ...ratorname{Aut}(P)[/itex] given as the group of [[stability automorphism]]s of this subnormal series, i.e., as follows:
5 KB (868 words) - 06:27, 25 November 2012
• {{coprime automorphism-faithful subgroup statement}} ...dd-order theorem by [[Walter Feit]] and [[John Griggs Thompson]]. The part of the paper containing this theorem (Chapter 2, Lemma 8.2, see also the [[#re
16 KB (2,524 words) - 20:27, 6 December 2011
• connective = of}} ...th>, sometimes called [itex]D_4[/itex], also called the {{dihedral group}} of order eight or the dihedral group acting on four elements, is defined by th
23 KB (3,126 words) - 19:49, 7 June 2012
• ...math> is a vector space over the [[prime field]] [itex]\mathbb{F}_p[/itex] of dimension greater than [itex]1[/itex]. In other words, [itex]V[/itex] is an ...V)[/itex] (the [[general linear group]] on [itex]V[/itex]), there exists a finite [itex]p[/itex]-group [itex]P[/itex] such that [itex]P/\Phi(P) \cong V</math
1 KB (209 words) - 00:42, 24 April 2009
• connective = of}} ...uaternion group]] and more on the element structure at [[element structure of quaternion group]].
11 KB (1,452 words) - 20:04, 29 June 2011
• # There exists exactly one isomorphism class of groups of that [[order of a group|order]]. # Any group of that order is a [[cyclic group]].
8 KB (1,323 words) - 16:45, 24 July 2017
• {{coprime automorphism group statement}} ...s termed [itex]H[/itex]-invariant if it equals its image under any element of [itex]H[/itex].
4 KB (692 words) - 15:30, 11 March 2009
• {{automorphism group control result}} ...group is either the [[trivial group]] or [[cyclic group:Z2|a cyclic group of order two]].
2 KB (289 words) - 00:35, 21 June 2013
• ...hen, [itex]G[/itex] is a [[fact about::metacyclic group]]: it has a [[fact about::cyclic normal subgroup]] such that the quotient is also a cyclic group. ...lly on it, and hence, the quotient is a subgroup of the automorphism group of the original cyclic normal subgroup.
5 KB (783 words) - 23:09, 3 March 2009
• ...strained]]: the [[normalizer]] of any non-identity [itex]p[/itex]-subgroup of [itex]G[/itex] is a [itex]p[/itex]-constrained group. ...th>P[/itex], not in [itex]G[/itex]) and further, [itex]A[/itex] has [[rank of a p-group|rank]] at least three. In other words, any generating set for <ma
9 KB (1,400 words) - 06:00, 30 July 2013
• stronger = isomorph-normal coprime automorphism-invariant subgroup| group property = group of prime power order}}
2 KB (233 words) - 19:15, 7 August 2009
• ...order eight]] and the cyclic maximal subgroup (the subgroup of order four) of this group. We call the dihedral group [itex]G[/itex], and use the followin In the typical description of [itex]G[/itex] as a permutation group:
20 KB (2,636 words) - 02:48, 6 July 2019
• ...p:D8|details on the subgroup structure]]) and the two Klein four-subgroups of this group. We call the dihedral group [itex]G[/itex], and use the followin ...subgroup]]s. Each of them has two cosets: the subgroup itself and the rest of the group.
14 KB (1,926 words) - 21:15, 12 August 2013
• ...[alternating group:A4|alternating group of degree four]], i.e., the subset of [itex]G[/itex] comprising the [[even permutation]]s. ...unique other coset (which is both a left coset and right coset) is the set of odd permutations.
9 KB (1,218 words) - 19:07, 30 April 2012
• ...ince it belongs to the family of [[standard representation]]s of symmetric groups. | Degree of representation || 2
14 KB (1,983 words) - 05:50, 7 June 2012