Perfect group

Remarks

Equivalently, the abelianisation $G/[G, G]$ is trivial, that is, every homomorphism from $G$ to an abelian group is trivial. Example: the alternating group $A_5$ is perfect, since $[A_5, A_5]$ is a nontrivial normal subgroup (as $A_5$ is not abelian) and $A_5$ is simple. Non-example: the symmetric group $S_n$ for $n \ge 2$ is not perfect, since the sign map $S_n \to \set{\pm 1}$ is a nontrivial homomorphism to an abelian group, so $[S_n, S_n] \subseteq A_n \ne S_n$. Every non-abelian simple group is perfect, by the same argument as for $A_5$. For a ring $R$ with identity, the elementary group $E(R)$ is perfect, since $e_{ij}(r) = [e_{ik}(r), e_{kj}(1)]$ for distinct $i, j, k$.