Commutator subgroup
The commutator subgroup
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 .