Perfect group

In mathematics, in the realm of group theory, a group is said to be perfect if it equals its own commutator subgroup.

The smallest (non-trivial) perfect group is the alternating group A₅. More generally, any non-abelian simple group is perfect since the commutator subgroup is a normal subgroup with abelian quotient. Of course a perfect group need not be simple, as the special linear group SL(2,5) (or the binary icosahedral group which is isomorphic to it) is an example of a perfect extension of the projective special linear group PSL(2,5) (which is isomorphic to A₅). A non-trivial perfect group, however, is necessarily not solvable.

Every acyclic group is perfect, but the converse is not true: ^[1] A₅ is perfect but not acyclic (in fact, not even superperfect).

[edit] Grün's lemma

A basic fact about perfect groups is Grün's lemma: the quotient of a perfect group by its center is centerless (has trivial center).^[2]

I.e., if Z(G) denotes the center of a given group G, and G is perfect, then the center of the quotient group G ⁄ Z(G) is the trivial group:

$G \mbox{ perfect} \implies Z \left( \frac{G}{Z(G)} \right) \cong \{1\}.$

As consequence, all higher centers of a perfect group equal the center.

^ A. Jon Berrick and Jonathan A. Hillman, "Perfect and acyclic subgroups of finitely presentable groups", Journal of the London Mathematical Society (2) 68 (2003), no. 3, 683--698. MR 2009444
^ Rose, John S. (1994). A Course in Group Theory. New York: Dover Publications, Inc., 61. ISBN 0-486-68194-7. MR 1298629