Derived series

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 ${\displaystyle [g,h]=g^{-1}h^{-1}gh}$ (or ${\displaystyle [g,h]=ghg^{-1}h^{-1}}$). The commutator ${\displaystyle [g,h]}$ is equal to the identity element e if and only if ${\displaystyle hg=gh}$ , that is, if and only if g and h commute. In general, ${\displaystyle gh=hg[g,h]}$ (or, respectively, ${\displaystyle gh=[g,h]hg}$).

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