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Group theory:)

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

[a,b] = a^{-1}b^{-1}ab;

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.