Perfect group

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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, or equivalently, if the group has no nontrivial abelian quotients.

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), see (Berrick & Hillman 2003).

Contents 1 Grün's lemma 2 Quasi-perfect group 3 References 4 External links

[edit] Grün's lemma

A basic fact about perfect groups is Grün's lemma from (Grün 1935, Satz 4): the quotient of a perfect group by its center is centerless (has trivial center).

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:

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

[edit] Quasi-perfect group

Especially in the field of algebraic K-theory, a group is said to be quasi-perfect if its commutator subgroup is perfect, (Karoubi 1973, pp. 301–411) and (Inassaridze 1995, p. 76).

[edit] 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. MR2009444
• Grün, Otto (1935), "Beiträge zur Gruppentheorie. I." (in German), Journal für Reine und Angewandte Mathematik 174: 1–14, Zbl 0012.34102, ISSN 0075-4102, http://www.digizeitschriften.de/index.php?id=loader&tx_jkDigiTools_pi1%5BIDDOC%5D=505862 
• Inassaridze, Hvedri (1995), Algebraic K-theory, Mathematics and its Applications, 311, Dordrecht: Kluwer Academic Publishers Group, MR1368402, ISBN 978-0-7923-3185-8, http://books.google.com/books?id=rnSE3aoNVY0C 
• Karoubi, M.: Périodicité de la K-théorie hermitienne, Hermitian K-Theory and Geometric Applications, Lecture Notes in Math. 343, Springer-Verlag, 1973
• Rose, John S. (1994). A Course in Group Theory. New York: Dover Publications, Inc.. pp. 61. MR1298629. ISBN 0-486-68194-7. 

[edit] External links

• Weistein, Eric W., "Perfect Group" from MathWorld.
• Weistein, Eric W., "Grün's lemma" from MathWorld.