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 
.
