Perfect group

2025-04-07T22:44:06Z

2600:1700:7F82:1000:C422:ED31:2E19:1AB4: Correctly correcting my error introduced in the first edit and incorrectly corrected in the second, sorry ….

{{short description|Mathematical group with trivial abelianization}}
In [[mathematics]], more specifically in [[group theory]], a [[Group (mathematics)|group]] is said to be '''perfect''' if it equals its own [[commutator subgroup]], or equivalently, if the group has no [[trivial group|non-trivial]] [[abelian group|abelian]] [[quotient group|quotients]].

== Examples ==
The smallest (non-trivial) perfect group is the [[alternating group]] ''A''5. More generally, any [[non-abelian group|non-abelian]] [[simple group]] is perfect since the commutator subgroup is a [[normal subgroup]] with abelian quotient. However, a perfect group need not be simple; for example, the [[special linear group]] over the [[field (mathematics)|field]] with 5 elements, SL(2,5) (or the [[binary icosahedral group]], which is [[group isomorphism|isomorphic]] to it) is perfect but not simple (it has a non-trivial [[center (group)|center]] containing <math>-\!\left(\begin{smallmatrix}1 & 0 \\ 0 & 1\end{smallmatrix}\right) = \left(\begin{smallmatrix}4 & 0 \\ 0 & 4\end{smallmatrix}\right)</math>).

The [[Direct product of groups|direct product]] of any two simple non-abelian groups is perfect but not simple; the commutator of two elements is [(''a'',''b''),(''c'',''d'')] = ([''a'',''c''],[''b'',''d'']). Since commutators in each simple group form a generating set, pairs of commutators form a generating set of the direct product.

The fundamental group of <math>SO(3)/I_{60}</math> is a perfect group of order 120.<ref>Milnor, John. "The Poincaré Conjecture." ''The millennium prize problems'' (2006): 70.</ref>

More generally, a [[quasisimple group]] (a perfect [[Central extension (mathematics)|central extension]] of a simple group) that is a non-trivial extension (and therefore not a simple group itself) is perfect but not simple; this includes all the [[soluble group|insoluble]] non-simple finite special linear groups SL(''n'',''q'') as extensions of the [[projective special linear group]] PSL(''n'',''q'') (SL(2,5) is an extension of PSL(2,5), which is isomorphic to ''A''5). Similarly, the special linear group over the [[real number|real]] and [[complex number|complex]] numbers is perfect, but the general linear group GL is never perfect (except when trivial or over <math>\mathbb{F}_2</math>, where it equals the special linear group), as the [[determinant]] gives a non-trivial abelianization and indeed the commutator subgroup is SL.

A non-trivial perfect group, however, is necessarily not [[solvable group|solvable]]; and 4 [[divisor|divides]] its [[order (group theory)|order]] (if finite), moreover, if 8 does not divide the order, then 3 does.<ref>Tobias Kildetoft (7 July 2015), [https://math.stackexchange.com/a/1357886/330413 answer] to [https://math.stackexchange.com/q/1357885/330413 "Is a non-trivial finite perfect group of order 4n?"]. ''Mathematics [[StackExchange]]''. Accessed 7 July 2015.</ref>

Every [[acyclic group]] is perfect, but the converse is not true: ''A''5 is perfect but not acyclic (in fact, not even [[Superperfect group|superperfect]]), see {{harv|Berrick|Hillman|2003}}. In fact, for <math>n\ge 5</math> the alternating group <math>A_n</math> is perfect but not superperfect, with <math>H_2(A_n,\Z) = \Z/2</math> for <math>n \ge 8</math>.

Any quotient of a perfect group is perfect. A non-trivial finite perfect group that is not simple must then be an extension of at least one smaller simple non-abelian group. But it can be the extension of more than one simple group. In fact, the direct product of perfect groups is also perfect.

Every perfect group ''G'' determines another perfect group ''E'' (its [[universal central extension]]) together with a [[surjection]] ''f'': ''E'' → ''G'' whose [[kernel (algebra)|kernel]] is in the center of ''E,''
such that ''f'' is universal with this property. The kernel of ''f'' is called the [[Schur multiplier]] of ''G'' because it was first studied by [[Issai Schur]] in 1904; it is isomorphic to the [[homology group]] <math>H_2(G)</math>.

In the '''plus construction''' of [[algebraic K-theory]], if we consider the group <math>\operatorname{GL}(A) = \text{colim} \operatorname{GL}_n(A)</math> for a [[commutative ring]] <math>A</math>, then the [[subgroup]] of elementary matrices <math>E(R)</math> forms a perfect subgroup.

== Ore's conjecture ==
As the commutator subgroup is ''generated'' by commutators, a perfect group may contain elements that are products of commutators but not themselves commutators. [[Øystein Ore]] showed in 1951 that the alternating groups on five or more elements contained only commutators, and [[conjecture]]d that this was so for all the finite non-abelian simple groups. Ore's conjecture was finally [[mathematical proof|proven]] in 2008. The proof relies on the [[classification of finite simple groups|classification theorem]].<ref>{{cite journal|authorlink1=Martin Liebeck |last1=Liebeck |first1=Martin |last2=O'Brien |first2=E.A. |last3=Shalev |first3=Aner |authorlink3=Aner Shalev |last4=Tiep |first4=Pham Huu |authorlink4=Pham Huu Tiep |title=The Ore conjecture|url=https://www.math.auckland.ac.nz/~obrien/research/ore.pdf|journal=[[Journal of the European Mathematical Society ]] |volume=12|year=2010|issue=4 |pages=939–1008|doi=10.4171/JEMS/220 |doi-access=free}}</ref>

==Grün's lemma==
A basic fact about perfect groups is '''Grün's lemma''' {{harv|Grün|1935|loc=Satz 4,<ref group="note">''[[wikt:Satz#German|Satz]]'' is German for "theorem".</ref> p. 3}}, due to [[Otto Grün]]: the [[quotient group|quotient]] of a perfect group by its [[center (group theory)|center]] is centerless (has trivial center).

<blockquote>'''Proof:''' If ''G'' is a perfect group, let ''Z''1 and ''Z''2 denote the first two terms of the [[Central series#Upper central series|upper central series]] of ''G'' (i.e., ''Z''1 is the center of ''G'', and ''Z''2/''Z''1 is the center of ''G''/''Z''1). If ''H'' and ''K'' are subgroups of ''G'', denote the [[commutator]] of ''H'' and ''K'' by [''H'', ''K''] and note that [''Z''1, ''G''] = 1 and [''Z''2, ''G''] ⊆ ''Z''1, and consequently (the convention that [''X'', ''Y'', ''Z''] = [[''X'', ''Y''], ''Z''] is followed):

:<math>[Z_2,G,G]=[[Z_2,G],G]\subseteq [Z_1,G]=1</math>
:<math>[G,Z_2,G]=[[G,Z_2],G]=[[Z_2,G],G]\subseteq [Z_1,G]=1.</math>

By the [[three subgroups lemma]] (or equivalently, by the [[Commutator#Identities (group theory)|Hall-Witt identity]]), it follows that [''G'', ''Z''2] = [[''G'', ''G''], ''Z''2] = [''G'', ''G'', ''Z''2] = {1}. Therefore, ''Z''2 ⊆ ''Z''1 = ''Z''(''G''), and the center of the quotient group ''G'' / ''Z''(''G'') is the [[trivial group]].</blockquote>

As a consequence, all [[Center (group theory)#Higher centers|higher centers]] (that is, higher terms in the [[upper central series]]) of a perfect group equal the center.

==Group homology==
In terms of [[group homology]], a perfect group is precisely one whose first homology group vanishes: ''H''1(''G'', '''Z''') = 0, as the first homology group of a group is exactly the abelianization of the group, and perfect means trivial abelianization. An advantage of this definition is that it admits strengthening:
* A [[superperfect group]] is one whose first two homology groups vanish: <math>H_1(G,\Z)=H_2(G,\Z)=0</math>.
* An [[acyclic group]] is one ''all'' of whose (reduced) homology groups vanish <math>\tilde H_i(G;\Z) = 0.</math> (This is equivalent to all homology groups other than <math>H_0</math> vanishing.)

==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; in symbols, a quasi-perfect group is one such that ''G''(1) = ''G''(2) (the commutator of the commutator subgroup is the commutator subgroup), while a perfect group is one such that ''G''(1) = ''G'' (the commutator subgroup is the whole group). See {{harv|Karoubi|1973|pp=301–411}} and {{harv| Inassaridze | 1995 | p=76}}.

==Notes==
{{reflist | group = note }}

==References==
{{reflist}}
{{refbegin}}
* {{Citation| first1= A. Jon|last1=Berrick|first2=Jonathan A.|last2=Hillman|title=Perfect and acyclic subgroups of finitely presentable groups|journal=[[London Mathematical Society|Journal of the London Mathematical Society]] |series=Second Series|volume=68|year=2003|number=3|pages=683–98|mr=2009444|doi=10.1112/s0024610703004587|s2cid=30232002 }}
* {{Citation | last1=Grün | first1=Otto | title=Beiträge zur Gruppentheorie. I. | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002173409 | language=German | zbl=0012.34102 | year=1935 | journal=[[Crelle's Journal|Journal für die Reine und Angewandte Mathematik]] | issn=0075-4102 | volume=174 | pages=1–14}}
*{{Citation | last1=Inassaridze | first1=Hvedri | title=Algebraic K-theory | url=https://books.google.com/books?id=rnSE3aoNVY0C | publisher=Kluwer Academic Publishers Group | location=Dordrecht | series=Mathematics and its Applications | isbn=978-0-7923-3185-8 | mr=1368402 | year=1995 | volume=311}}
* {{Citation|last=Karoubi|first=Max|title=Périodicité de la K-théorie hermitienne, Hermitian K-Theory and Geometric Applications|series= Lecture Notes in Math. |volume=343|publisher=Springer-Verlag|year=1973}}
*{{Citation
| 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
| mr = 1298629
}}
{{refend}}

==External links==
* {{MathWorld|urlname=PerfectGroup|title=Perfect Group}}
* {{MathWorld|urlname=GruensLemma|title=Grün's lemma}}

[[Category:Properties of groups]]
[[Category:Lemmas]]

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Perfect group