The commutator subgroup (also called a derived group) of a group is the subgroup generated by the commutators of its elements, and is commonly denoted or . It is the unique smallest normal subgroup of such that is Abelian (Rose 1994, p. 59). It can range from the identity subgroup (in the case of an Abelian group) to the whole group. Note that not every element of the commutator subgroup is necessarily a commutator.
For instance, in the quaternion group (, , , ) with eight elements, the commutators form the subgroup . The commutator subgroup of the symmetric group is the alternating group. The commutator subgroup of the alternating group is the whole group . When , is a simple group and its only nontrivial normal subgroup is itself. Since is a nontrivial normal subgroup, it must be .
The first homology of a group is the Abelianization