The smallest (non-trivial) perfect group is the
alternating group"A"5. More generally, any non-abelian simple groupis perfect since the commutator subgroup is a normal subgroupwith abelian quotient. Of course a perfect group need not be simple, as the special linear group"SL"(2,5) (or the binary icosahedral groupwhich is isomorphic to it) is an example of a perfect extension of the projective special linear group"PSL"(2,5) (which is isomorphic to "A"5). A non-trivial perfect group, however, is necessarily not solvable.
Every acyclic group is perfect, but the converse is not true: [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. MathSciNet|id=2009444] "A"5 is perfect but not acyclic (in fact, not even superperfect).
A basic fact about perfect groups is Grün's lemma: the quotient of a perfect group by its center is centerless (has trivial center). [cite book
last = Rose
first = John S.
title = A Course in Group Theory
publisher = Dover Publications, Inc.
location = New York
pages = 61
year = 1994
isbn = 0-486-68194-7 MathSciNet|id=1298629]
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
As consequence, all higher centers of a perfect group equal the center.
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