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- ...uotient of the [[free group]] on <math>d</math> generators by the [[normal subgroup]] generated by all <math>n^{th}</math> 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 ...] <math>H</math> of a [[group]] <math>G</matH> is termed a fully invariant subgroup of <math>G</math> 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 <math>F</math> and a verbal subgroup <math>V</math> of <math>F</math> such that <math>G \cong F/V</math> ==Relation with other properties==3 KB (480 words) - 22:55, 29 June 2010
- ...] <math>p</math>. We give equivalent definitions of a <math>p</math>-Sylow subgroup. ...ubgroup <math>H</math> of a group <math>G</math> is a <math>p</math>-Sylow subgroup of <math>G</math> if ... !! Applications to... !! Additional comments16 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 twenti16 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 <math>\subseteq</math>. When talking of [[subgroup]]s, we typically use <math>\le</math> 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 a22 KB (3,451 words) - 10:45, 4 April 2013
- * 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
- {{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