# Search results

• 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). || $G$ equals the derived subgroup $[G,G]$, sometimes also denoted $G'$. ...d as a product in the group of finitely many elements each of which is a [[commutator]]. || for any $g \in G$, there exist elements $g_1,g_2,\dot 7 KB (997 words) - 03:09, 11 February 2013 • A [[group]] is said to be '''Schur-trivial''' or a '''group with trivial Schur multiplier''' if it satisfies the following e ...oup]] defined by the commutator map is an [[isomorphism of groups]] to the derived subgroup. 5 KB (641 words) - 08:39, 25 February 2013 • ! Type of group !! High occurrence or low occurrence? !! Some or all characteristic subgroups !! Explanation/comment ...her than trivial subgroup and whole group || The additive group of a field or of a vector space over a field is a [[group whose automorphism group is tra 40 KB (4,935 words) - 13:17, 2 June 2020 • .../math>) is contained in [itex]H$. || [[proving normality#Compute its commutator with the whole group|proving normality]] || ...In words, we say that $\! H$ is ''normal in'' $\! G$ or a ''normal subgroup of'' $\! G$.
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 $n \ge 3$, $SL_n(k)$ is a [[perfect group]] for any field $k$.
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]] $SL_n(k)$ 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