A group that coincides with its commutator subgroup.
If is a non-Abelian group, its commutator subgroup is a normal subgroup other than the trivial group. It follows that if is simple, it must be perfect. The converse, however, is not necessarily true. For example, the special linear group is always perfect if (Rose 1994, p. 61), but if is not a power of 2 (i.e., the field characteristic of the finite field is not 2), it is not simple, since its group center contains two elements: the identity matrix and its additive inverse , which are different because .