Commutator Subgroup - Commutators

Commutators

For elements g and h of a group G, the commutator of g and h is . The commutator is equal to the identity element e if and only if gh = hg, that is, if and only if g and h commute. In general, gh = hg.

An element of G which is of the form for some g and h is called a commutator. The identity element e = is always a commutator, and it is the only commutator if and only if G is abelian.

Here are some simple but useful commutator identities, true for any elements s, g, h of a group G:

• , where .
• For any homomorphism f: G → H, f = .

The first and second identities imply that the set of commutators in G is closed under inversion and under conjugation. If in the third identity we take H = G, we get that the set of commutators is stable under any endomorphism of G. This is in fact a generalization of the second identity, since we can take f to be the conjugation automorphism .

However, the product of two or more commutators need not be a commutator. A generic example is in the free group on a,b,c,d. It is known that the least order of a finite group for which there exists two commutators whose product is not a commutator is 96; in fact there are two nonisomorphic groups of order 96 with this property.