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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 $a$) and has four ''reflections'' each being an [[i The group is (up to isomorphism) the subgroup of the symmetric group on $\{ 1,2,3,4 \}$ given by:
19 KB (2,660 words) - 13:15, 14 February 2015
• ...|| It is possible to have a finitely generated group $G$ and a subgroup $H$ of $G$ such that $H$ is not finitely g ...th>G[/itex] is a finitely generated group and $H$ is a [[normal subgroup]] of $G$, then the [[quotient group]] $G/H$ 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]] $H$ of a [[group]] $G$ 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 $G/Z^{i-1}(G)$, and $Z^0(G)$ is the trivial subgroup. ...s [[lower central series]] stabilizes after a finite length at the trivial subgroup || there is a nonnegative integer $c$ such that $[[[..[G,G] 18 KB (2,458 words) - 23:24, 9 September 2016 • ...f a group [itex]G$ is pronormal if... (right-action convention) !! A subgroup $H$ of a group $G$ is pronormal if... (left-action co ...such that $H^g = H^x$. || for any $g \in G$ and any subgroup $K$ of $G$ containing both $H$ and $g 20 KB (2,702 words) - 04:06, 10 November 2014 • ...rase|prime number among groups, group without any proper nontrivial normal subgroup, group without any proper nontrivial quotients}} ...nt::normal subgroup]] || [itex]G$ is nontrivial and for any [[normal subgroup]] $H$ of $G$, either $H$ 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[/itex] is a spherical von Dyck group, i.e., it occurs as a finite subgroup of $SO(3,\R)$. 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]] $G' = [G,G]$ is trivial. ...p $\{ (g,g) \mid g \in G \}$ is a [[defining ingredient::normal subgroup]] inside $G \times G$.
13 KB (1,912 words) - 15:35, 11 April 2017
• Below are many '''equivalent''' definitions of characteristic subgroup. ...bgroup $H$ of a group $G$ is called a characteristic subgroup of $G$ 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 $H$ of a group $G$ is termed automorph-conjugate or i ...up), is also [[Defining ingredient::conjugate subgroups|conjugate]] to the subgroup. || for any $\sigma \in \operatorname{Aut}(G)$, 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 $G$ as a [[normal subgroup]] of some group $K$, $G$ 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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