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• ...uotient of the [[free group]] on $d$ generators by the [[normal subgroup]] generated by all $n^{th}$ powers. A Burnside group is a group ...particular, any Burnside group is a [[group in which every fully invariant subgroup is verbal]].
4 KB (586 words) - 05:01, 11 April 2017
• ==The three main formulations and their resolutions for different group properties== An inner automorphism of a group satisfies all these properties.
9 KB (1,273 words) - 17:43, 5 December 2015
• weaker = group with finite derived subgroup}} ...group]]. In fact, there is an explicit bound on the size of the commutator subgroup as a function of the size of the inner automorphism group.
4 KB (663 words) - 17:11, 29 June 2013
• ==Relation with other properties== ===Stronger properties===
10 KB (1,615 words) - 02:00, 28 June 2013
• A [[subgroup]] of a [[group]] is termed '''fully invariant''' or '''fully characteristic ...] $H$ of a [[group]] $G</matH> is termed a fully invariant subgroup of [itex]G$ if ...
17 KB (2,174 words) - 18:06, 12 August 2013
• ...oups. Analogous results hold when we restrict to groups satisfying certain properties. ...oups. Analogous results hold when we restrict to groups satisfying certain properties.
10 KB (1,456 words) - 18:49, 14 January 2014
• ...e that this definition differs completely from the definition of [[p-local subgroup]]. ==Conjunction with group properties==
5 KB (765 words) - 19:13, 1 June 2016
• ...ism]] of a group is termed '''normal''' or '''quotientable''' or '''normal subgroup-preserving''' if it satisfies the following equivalent conditions: # It takes each [[defining ingredient::normal subgroup]] to itself (bijectively).
3 KB (401 words) - 12:18, 30 June 2009
• ...nt::verbal subgroup]] || there is a free group $F$ and a verbal subgroup $V$ of $F$ such that $G \cong F/V$ ==Relation with other properties==
3 KB (480 words) - 22:55, 29 June 2010
• ...] $p$. We give equivalent definitions of a $p$-Sylow subgroup. ...ubgroup $H$ of a group $G$ is a $p$-Sylow subgroup of $G$ if ... !! Applications to... !! Additional comments
16 KB (2,017 words) - 19:02, 20 December 2014
• {{survey article|normal subgroup}} ...In this article, we look at the many reasons why normality is an important subgroup property, and why it keeps popping up repeatedly.
16 KB (2,588 words) - 04:43, 7 January 2010
• ...e it the zero element in any group is a subgroup of it (namely the trivial subgroup). The analogous statement is not true for many other algebraic varieties -- ==Properties==
4 KB (578 words) - 20:09, 29 July 2013
• {{variational survey article|characteristic subgroup}} Characteristicity is a pivotal [[subgroup property]], that, along with [[normality]], dates back to before the twenti
16 KB (2,240 words) - 21:17, 22 March 2010
• {{subgroup property related to|geometric group theory}} {{subgroup property related to|combinatorial group theory}}
10 KB (1,445 words) - 17:20, 26 July 2013
• ...subset inclusions are denoted by $\subseteq$. When talking of [[subgroup]]s, we typically use $\le$ to emphasize that the subset also has ...sequent work by others led to notions (e.g., [[solvable group]], [[normal subgroup]]) that would have been difficult to develop by viewing a group merely as a
22 KB (3,451 words) - 10:45, 4 April 2013
• * 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
• {{subgroup metaproperty satisfaction}} ...]] satisfies the [[subgroup metaproperty]] of being [[strongly join-closed subgroup property|strongly join-closed]].
8 KB (1,202 words) - 16:54, 24 February 2011
• {{subgroup metaproperty satisfaction| property = normal subgroup|
10 KB (1,652 words) - 16:50, 24 February 2011

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