Editing Commutator subgroup (section)

== Examples ==
* The commutator subgroup of any [[abelian group]] is [[Trivial group|trivial]].
* The commutator subgroup of the [[general linear group]] <math>\operatorname{GL}_n(k)</math> over a [[Field (mathematics)|field]] or a [[division ring]] ''k'' equals the [[special linear group]] <math>\operatorname{SL}_n(k)</math> provided that <math>n \ne 2</math> or ''k'' is not the [[finite field|field with two elements]].<ref>{{citation|author=Suprunenko|first=D.A.|title=Matrix groups|publisher=American Mathematical Society|year=1976|series=Translations of Mathematical Monographs}}, Theorem II.9.4</ref>
* The commutator subgroup of the [[alternating group]] ''A''<sub>4</sub> is the [[Klein four group]].
* The commutator subgroup of the [[symmetric group]] ''S<sub>n</sub>'' is the [[alternating group]] ''A<sub>n</sub>''.
* The commutator subgroup of the [[quaternion group]] ''Q'' = {1, &minus;1, ''i'', &minus;''i'', ''j'', &minus;''j'', ''k'', &minus;''k''} is [''Q'',''Q''] = {1, &minus;1}.

=== Map from Out ===
Since the derived subgroup is [[Characteristic subgroup|characteristic]], any automorphism of ''G'' induces an automorphism of the abelianization. Since the abelianization is abelian, [[inner automorphism]]s act trivially, hence this yields a map
:<math>\operatorname{Out}(G) \to \operatorname{Aut}(G^{\mbox{ab}})</math>