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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: A₅ 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 centre has trivial centre.

References

• 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.