commutator subgroup
Every group has a precise measure of “how nonabelian” it is, packaged into a single subgroup. The commutator subgroup collects all the failures of commutativity — the elements of the form ghg^{-1}h^{-1} that would be the identity if everything commuted. Crushing the group by exactly this subgroup produces the best possible abelian approximation of the group.
Formally, the commutator of g and h is [g, h] = g h g^{-1} h^{-1}, and the commutator subgroup (or derived subgroup) G' = [G, G] is the subgroup generated by all such commutators. It is a normal — indeed characteristic — subgroup, and the quotient G/G' is abelian. Crucially G' is the smallest normal subgroup with abelian quotient: G/N is abelian if and only if N contains G'. For this reason G/G' is called the abelianization of G, written G^{ab}.
The abelianization is the universal abelian quotient: every homomorphism from G to an abelian group factors uniquely through G^{ab}. Iterating the construction — G, then G', then G'' = (G')', and so on — gives the derived series, whose termination at the trivial group is the very definition of solvability. A small warning: the commutator subgroup is generated by commutators but need not consist only of commutators; in some groups a product of commutators is not itself a single commutator.
For S_n with n ≥ 2 the commutator subgroup is the alternating group A_n, so the abelianization S_n^{ab} is Z/2Z — recording only the sign of a permutation. For the nonabelian dihedral group D_n with n odd, the commutator subgroup is the full rotation subgroup, and the abelianization is Z/2Z.
The commutator subgroup of S_n is A_n; its abelianization is Z/2Z.