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  • {{subgroup-defining function}} ...' of a group is the set of its central elements. The center is clearly a [[subgroup]].
    15 KB (2,081 words) - 20:14, 1 June 2016
  • ...gon). This has a cyclic subgroup comprising rotations (which is the cyclic subgroup generated by <math>a</math>) and has four ''reflections'' each being an [[i The group is (up to isomorphism) the subgroup of the symmetric group on <math>\{ 1,2,3,4 \}</math> given by:
    19 KB (2,660 words) - 13:15, 14 February 2015
  • ...|| It is possible to have a finitely generated group <math>G</math> and a subgroup <math>H</math> of <math>G</math> such that <math>H</math> is not finitely g ...th>G</math> is a finitely generated group and <math>H</math> is a [[normal subgroup]] of <math>G</math>, then the [[quotient group]] <math>G/H</math> is a fini
    7 KB (920 words) - 03:15, 16 April 2017
  • {{subgroup-defining function}} ...tion of all maximal subgroups, biggest subgroup contained in every maximal subgroup}}
    4 KB (603 words) - 06:49, 20 April 2017
  • ...subgroups]], [[quick phrase::no isomorphic copies]], [[quick phrase::only subgroup of its isomorphism type]]}} A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is said to be '''isomorph-fr
    11 KB (1,293 words) - 02:34, 20 April 2016
  • * It is the subgroup of [[symmetric group:S4|the symmetric group of degree four]] comprising the ==Group properties==
    5 KB (643 words) - 16:32, 21 December 2014
  • {{subgroup property}} ...] and the only normal subgroup properly contained inside it is the trivial subgroup.
    6 KB (941 words) - 05:30, 24 January 2015
  • ...math> to <math>G/Z^{i-1}(G)</math>, and <math>Z^0(G)</math> is the trivial subgroup. ...s [[lower central series]] stabilizes after a finite length at the trivial subgroup || there is a nonnegative integer <math>c</math> such that <math>[[[..[G,G]
    18 KB (2,458 words) - 23:24, 9 September 2016
  • ...f a group <math>G</math> is pronormal if... (right-action convention) !! A subgroup <math>H</math> of a group <math>G</math> is pronormal if... (left-action co ...such that <math>H^g = H^x</math>. || for any <math>g \in G</math> and any subgroup <math>K</math> of <math>G</math> containing both <math>H</math> and <math>g
    20 KB (2,702 words) - 04:06, 10 November 2014
  • ...rase|prime number among groups, group without any proper nontrivial normal subgroup, group without any proper nontrivial quotients}} ...nt::normal subgroup]] || <math>G</math> is nontrivial and for any [[normal subgroup]] <math>H</math> of <math>G</math>, either <math>H</math> is trivial or <ma
    10 KB (1,387 words) - 20:01, 23 April 2014
  • {{subgroup property}} A [[subgroup]] of a [[group]] is termed '''subnormal''' if any of the following equivale
    18 KB (2,510 words) - 15:40, 16 April 2017
  • | {{arithmetic function value given order|derived length|2|6}} || Cyclic subgroup of order three is abelian, has abelian quotient. | {{arithmetic function value given order|subgroup rank of a group|2|6}} || All proper subgroups are cyclic.
    18 KB (2,584 words) - 16:41, 18 January 2015
  • ...math>S_4</math> is a spherical von Dyck group, i.e., it occurs as a finite subgroup of <math>SO(3,\R)</math>. In particular, this makes it a [[Coxeter group]]. | {{arithmetic function value given order|subgroup rank of a group|2|24}} ||All proper subgroups are cyclic or have generating
    16 KB (2,214 words) - 17:29, 19 May 2014
  • * Its [[defining ingredient::derived subgroup]] <math>G' = [G,G]</math> is trivial. ...p <math>\{ (g,g) \mid g \in G \}</math> is a [[defining ingredient::normal subgroup]] inside <math>G \times G</math>.
    13 KB (1,912 words) - 15:35, 11 April 2017
  • Below are many '''equivalent''' definitions of characteristic subgroup. ...bgroup <math>H</math> of a group <math>G</math> is called a characteristic subgroup of <math>G</math> if ...
    40 KB (4,850 words) - 00:04, 18 March 2019
  • ...ight cosets]], [[quick phrase::kernel of a homomorphism]], [[quick phrase::subgroup that is a union of conjugacy classes]]}} ...g any of these definitions, we first need to check that we actually have a subgroup.
    43 KB (5,764 words) - 13:39, 2 August 2018
  • ...d automorph-conjugate or intravariant if... (right-action convention) !! A subgroup <math>H</math> of a group <math>G</math> is termed automorph-conjugate or i ...up), is also [[Defining ingredient::conjugate subgroups|conjugate]] to the subgroup. || for any <math>\sigma \in \operatorname{Aut}(G)</math>, there exists <ma
    14 KB (1,758 words) - 17:39, 21 December 2014
  • {{subgroup metaproperty dissatisfaction| property = normal subgroup|
    16 KB (2,520 words) - 05:52, 15 October 2013
  • * Whenever it is embedded as a [[normal subgroup]] inside a bigger group, it is actually a [[direct factor]] inside that big * For any embedding of <math>G</math> as a [[normal subgroup]] of some group <math>K</math>, <math>G</math> is a [[direct factor]] of <m
    3 KB (388 words) - 05:07, 9 October 2015
  • ...lue|subgroup rank of a group|1}} || The group is cyclic, hence so is every subgroup. ==Group properties==
    3 KB (344 words) - 16:16, 29 April 2014

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