Commutator subgroup

The commutator subgroup is a normal subgroup of generated by the group commutator of all elements, pairwise. #m/def/group

Proof of normal subgroup

is a subgroup by construction. Let . Then for any conjugate it follows , so and thus . Therefore is a normal subgroup.

Wikipedia notes

[the commutator subgroup] is stable under every endomorphism of : that is, is a fully characteristic subgroup of , a property considerably stronger than normality.

Properties

A quotient with the commutator subgroup of is called an Abelianization of .

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