# Perfect group

In

mathematics , in the realm ofgroup theory , a group is said to be**perfect**if it equals its owncommutator 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-abeliansimple group is perfect since the commutator subgroup is anormal subgroup with abelian quotient. Of course a perfect group need not be simple, as thespecial linear group "SL"(2,5) (or thebinary icosahedral group which is isomorphic to it) is an example of a perfect extension of theprojective 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=1298629I.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****External links***

*

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

**Almost perfect group**— In mathematics, a subgroup A of a group G is said to be almost perfect if every element a of A may be written as a product of commutators a g(a ) 1g 1, where a is in A and g is in G. This is intended to contrast a perfect group P, in which every… … Wikipedia**Perfect (disambiguation)**— Perfect may refer to: * Perfection, a philosophical concept * Perfection (law), a legal concept * Perfect aspect, a grammatical concept * Perfect, a Cathar priestIn mathematics: * Perfect number * Perfect group * Perfect set * Perfect graphIn… … Wikipedia**Perfect Strangers**— Студийный альбом Deep Purple Дата выпуска 2 ноября 1984 Записан … Википедия**Perfect Strangers (album)**— Perfect Strangers Studio album by Deep Purple Released 16 September 1984 … Wikipedia**Perfect Songs**— is a music publishing company founded in 1982 by Trevor Horn and his his wife Jill Sinclair. It is part of the SPZ group of companies owned by Horn and his associates, which also includes Sarm Studios (recording studios), Horn Productions (record … Wikipedia**Perfect fifth**— Play (help· … Wikipedia**Perfect Strangers (TV series)**— Perfect Strangers Perfect Strangers opening title from seasons 3 8. Format Sitcom Created by Dale McRaven … Wikipedia**Perfect Dark (series)**— Perfect Dark The original Perfect Dark logo Genres First person shooter, stealth, action … Wikipedia**Perfect Strangers**— Album par Deep Purple Sortie novembre 1984 Enregistrement août 1984 Durée 39:28 Genre hard rock Producteur … Wikipédia en Français**Perfect fourth**— Play (help· … Wikipedia