In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.
The commutator subgroup is important because it is the smallest normal subgroup such that the quotient group of the original group by this subgroup is abelian. In other words, G/N is abelian if and only if N contains the commutator subgroup. So in some sense it provides a measure of how far the group is from being abelian; the larger the commutator subgroup is, the "less abelian" the group is.
For elements g and h of a group G, the commutator of g and h is [g,h]: = g − 1h − 1gh. The commutator [g,h] 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[g,h].
An element of G which is of the form [g,h] for some g and h is called a commutator. The identity element e = [e,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:
- [g,h] − 1 = [h,g].
- [g,h]s = [gs,hs], where gs = s − 1gs.
For any homomorphism , f([g,h]) = [f(g),f(h)].
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 [a,b][c,d] 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.
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