(Group theory:)

The commutator of two elements a,b of a group is defined as

[a,b] = a-1b-1ab;
if a and b commute then [a,b]=1.

The subgroup generated by all commutators of the group is the commutator subgroup. Note the word "generated": the set of all commutators of the group is, generally, not a group (in any interesting case). While clumsily defined, the commutator subgroup is important.

An abelian group has only trivial commutators, as a⋅b⋅a-1⋅b-1=1. Hence its commutator subgroup is {1}. The converse is also (trivially) true.