perfect group

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

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

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

The alternating group A 5A_5 is the smallest nontrivial perfect group.

Created on May 24, 2010 at 21:07:39. See the history of this page for a list of all contributions to it.