Perfect group

In mathematics, in the realm of group theory, a group is said to be perfect if it equals its own commutator subgroup, or equivalently, if the group has no nontrivial abelian quotients.

The smallest (non-trivial) perfect group is the alternating group "A"₅. 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 "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).

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). [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 trivial group:

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

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

References

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