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  • ...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 <math>p</math>-Sylow subgroups divides the index of any <math>p</math>-Sylow subgroup and is also congruent to <math>1</math> 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 <math>P</math> be a finite <math>p</math>-group, i.e., a [[group of prime power order]] where the underlying prime is <math>p</math>. Suppose t ...ratorname{Aut}(P)</math> 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 <math>D_4</math>, 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]] <math>\mathbb{F}_p</math> of dimension greater than <math>1</math>. In other words, <math>V</math> is an ...V)</math> (the [[general linear group]] on <math>V</math>), there exists a finite <math>p</math>-group <math>P</math> such that <math>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 <math>H</math>-invariant if it equals its image under any element of <math>H</math>.
    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, <math>G</math> 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 <math>p</math>-subgroup of <math>G</math> is a <math>p</math>-constrained group. ...th>P</math>, not in <math>G</math>) and further, <math>A</math> 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 <math>G</math>, and use the followin In the typical description of <math>G</math> 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 <math>G</math>, 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 <math>G</math> 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