derived group

The commutator subgroup (also called a derived group) of a group G is the subgroup generated by the commutators of its elements, and is commonly denoted G' or [G, G]. It is the unique smallest normal subgroup of G such that G/[G, G] is Abelian. It can range from the identity subgroup (in the case of an Abelian group) to the whole group. Note that not every element of the commutator subgroup is necessarily a commutator.

Abelian group | Abelianization | commutator | group | normal subgroup | perfect group