# nLab perfect group

A group $G$ is perfect if it equals its own commutator subgroup $[G, G]$, i.e., if every element of $G$ is a product of commutators (elements of the form $[g, h] = g h g^{-1} h^{-1}$).

Equivalently: let $G^{ab}$ denote the abelianization of $G$ (the target of the homomorphism $G \to G^{ab}$ that is universal among homomorphisms from $G$ to abelian groups, or the largest abelian quotient of $G$). Then $G$ is perfect precisely when $G^{ab}$ is a trivial group, since

$G^{ab} \cong G/[G, G].$

The alternating group $A_5$ is the smallest nontrivial perfect group.

Created on May 24, 2010 21:07:39 by Todd Trimble (69.118.56.215)