Perfect Group - Examples

Examples

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. Conversely, a perfect group need not be simple; for example, the special linear group SL(2,5) (or the binary icosahedral group which is isomorphic to it) is perfect but not simple (it has a non-trivial center containing ).

More generally, a quasisimple group (a perfect central extension of a simple group) which is a non-trivial extension (i.e., not a simple group itself) is perfect but not simple; this includes all the insoluble non-simple finite special linear groups SL(n,q) as extensions of the projective special linear group PSL(n,q) (SL(2,5) is an extension of PSL(2,5), which is isomorphic to A₅). Similarly, the special linear group over the real and complex numbers is perfect, but the general linear group GL is never perfect (except when trivial or over F₂, where it equals the special linear group), as the determinant gives a non-trivial abelianization and indeed the commutator subgroup is SL.

A non-trivial perfect group, however, is necessarily not solvable.

Every acyclic group is perfect, but the converse is not true: A₅ is perfect but not acyclic (in fact, not even superperfect), see (Berrick & Hillman 2003). In fact, for n ≥ 5 the alternating group A_n is perfect but not superperfect, with H₂(A_n, Z) = Z/2 for n ≥ 8.