Perfect Group

A perfect group is a group $G$ such that $[G,G] = G,$ where $[G,G]$ denotes the commutator subgroup .

Equivalently, a group is perfect iff it has no nontrivial abelian quotient (its “abelianization” is trivial). In particular, no nontrivial abelian group is perfect, while every nonabelian simple group is perfect.

Examples:

• $A_n$ is perfect for $n\ge 5$ (in particular, $A_5$ is perfect).
• The trivial group $\{e\}$ is perfect.
• (Non-example) Any abelian group (e.g. $\mathbb{Z}$) is not perfect: its commutator subgroup is trivial.