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- The '''perfect core''' or '''stable commutator''' of a group is all of the following equivalent things: # The unique largest [[defining ingredient::perfect group|perfect]] [[subgroup]].2 KB (270 words) - 15:48, 6 February 2010
- ...(i.e., its commutator subgroup with itself). || <math>G</math> equals the derived subgroup <math>[G,G]</math>, sometimes also denoted <math>G'</math>. ...d as a product in the group of finitely many elements each of which is a [[commutator]]. || for any <math>g \in G</math>, there exist elements <math>g_1,g_2,\dot7 KB (997 words) - 03:09, 11 February 2013
- A [[group]] is said to be '''Schur-trivial''' or a '''group with trivial Schur multiplier''' if it satisfies the following e ...oup]] defined by the commutator map is an [[isomorphism of groups]] to the derived subgroup.5 KB (641 words) - 08:39, 25 February 2013
- ! Type of group !! High occurrence or low occurrence? !! Some or all characteristic subgroups !! Explanation/comment ...her than trivial subgroup and whole group || The additive group of a field or of a vector space over a field is a [[group whose automorphism group is tra40 KB (4,935 words) - 13:17, 2 June 2020
- .../math>) is contained in <math>H</math>. || [[proving normality#Compute its commutator with the whole group|proving normality]] || ...In words, we say that <math>\! H</math> is ''normal in'' <math>\! G</math> or a ''normal subgroup of'' <math>\! G</math>.43 KB (5,764 words) - 13:39, 2 August 2018
- The notion of derived subgroup or commutator subgroup naturally arose in the context of finding a natural choice for a g The '''derived subgroup''' or '''commutator subgroup''' of a [[group]] is defined in the following equivalent ways:6 KB (880 words) - 15:50, 8 July 2011
- property = perfect group}} * For <math>n \ge 3</math>, <math>SL_n(k)</math> is a [[perfect group]] for any field <math>k</math>.1 KB (185 words) - 22:39, 27 March 2013
- {{iterated series|commutator subgroup}} The '''derived series''' or '''commutator series''' of a [[group]] is defined as follows:3 KB (448 words) - 04:11, 17 February 2013
- ...r group]] <math>SL_n(k)</math> is a [[perfect group]]: it equals its own [[derived subgroup]]. # [[uses::Perfectness is quotient-closed]]: The quotient of a perfect group by a normal subgroup is perfect.2 KB (354 words) - 16:49, 15 May 2015
- ...[[coset]] of the [[defining ingredient::derived subgroup]] other than the commutator subgroup itself, forms exactly one [[defining ingredient::conjugacy class]] * [[Weaker than::Perfect group]]971 bytes (122 words) - 20:09, 20 May 2011