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| In [[mathematics]], more specifically in the area of [[Abstract algebra|modern algebra]] known as [[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 nontrivial [[abelian group|abelian]] [[quotient group|quotients]] (equivalently, its [[abelianization]], which is the universal abelian quotient, is trivial). In symbols, a perfect group is one such that ''G''<sup>(1)</sup> = ''G'' (the commutator subgroup equals the group), or equivalently one such that ''G''<sup>ab</sup> = {1} (its abelianization is trivial).
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| == Examples ==
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| The smallest (non-trivial) perfect group is the [[alternating group]] ''A''<sub>5</sub>. More generally, any non-[[abelian group|abelian]] [[simple group]] is perfect since the commutator subgroup is a [[normal subgroup]] with abelian quotient. Conversely, a perfect group need not be simple; for example, the [[special linear group]] SL(2,5) (or the [[binary icosahedral group]] which is 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>).
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| More generally, a [[quasisimple group]] (a perfect [[Central extension (mathematics)|central extension]] of a simple group) which is a non-trivial extension (i.e., not a simple group itself) is perfect but not simple; this includes all the 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''<sub>5</sub>). Similarly, the special linear group over the real and complex numbers is perfect, but the general linear group GL is never perfect (except when trivial or over '''F'''<sub>2</sub>, where it equals the special linear group), as the [[determinant]] gives a non-trivial abelianization and indeed the commutator subgroup is SL.
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| A non-trivial perfect group, however, is necessarily not [[solvable group|solvable]].
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| Every [[acyclic group]] is perfect, but the converse is not true: ''A''<sub>5</sub> is perfect but not acyclic (in fact, not even [[Superperfect group|superperfect]]), see {{harv|Berrick|Hillman|2003}}. In fact, for ''n'' ≥ 5 the alternating group ''A<sub>n</sub>'' is perfect but not superperfect, with ''H''<sub>2</sub>(''A<sub>n</sub>'', '''Z''') = '''Z'''/2 for ''n'' ≥ 8.
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| Every perfect group ''G'' determines another perfect group ''E'' (its [[universal central extension]]) together with a surjection ''f:E'' → ''G'' whose kernel is in the center of ''E,''
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| 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 [[Schur]] in 1904; it is isomorphic to the
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| homology group ''H<sub>2</sub>(G)''.
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| ==Grün's lemma==
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| A basic fact about perfect groups is '''Grün's lemma''' from {{harv|Grün|1935|loc=Satz 4,<ref group="note">''[[wikt:Satz#German|Satz]]'' is German for "theorem".</ref> p. 3}}: the [[quotient group|quotient]] of a perfect group by its [[center (group theory)|center]] is centerless (has trivial center).
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| <blockquote>'''Proof:''' If ''G'' is a perfect group, let ''Z''<sub>1</sub> and ''Z''<sub>2</sub> denote the first two terms of the [[Central_series#Upper_central_series|upper central series]] of ''G'' (i.e., ''Z''<sub>1</sub> is the center of ''G'', and ''Z''<sub>2</sub>/''Z''<sub>1</sub> is the center of ''G''/''Z''<sub>1</sub>). If ''H'' and ''K'' are subgroups of ''G'', denote the [[commutator]] of ''H'' and ''K'' by [''H'', ''K''] and note that [''Z''<sub>1</sub>, ''G''] = 1 and [''Z''<sub>2</sub>, ''G''] ⊆ ''Z''<sub>1</sub>, and consequently (the convention that [''X'', ''Y'', ''Z''] = [[''X'', ''Y''], ''Z''] is followed): | |
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| :<math>[Z_2,G,G]=[[Z_2,G],G]\subseteq [Z_1,G]=1</math>
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| :<math>[G,Z_2,G]=[[G,Z_2],G]=[[Z_2,G],G]\subseteq [Z_1,G]=1.</math>
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| By the [[three subgroups lemma]] (or equivalently, by the [[Commutator#Identities|Hall-Witt identity]]), it follows that [''G'', ''Z''<sub>2</sub>] = [[''G'', ''G''], ''Z''<sub>2</sub>] = [''G'', ''G'', ''Z''<sub>2</sub>] = {1}. Therefore, ''Z''<sub>2</sub> ⊆ ''Z''<sub>1</sub> = ''Z''(''G''), and the center of the quotient group ''G'' ⁄ ''Z''(''G'') is the [[trivial group]].</blockquote>
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| 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. | |
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| ==Group homology==
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| In terms of [[group homology]], a perfect group is precisely one whose first homology group vanishes: ''H''<sub>1</sub>(''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:
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| * A [[superperfect group]] is one whose first two homology groups vanish: ''H''<sub>1</sub>(''G'', '''Z''') = ''H''<sub>2</sub>(''G'', '''Z''') = 0.
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| * An [[acyclic group]] is one ''all'' of whose (reduced) homology groups vanish <math>\tilde H_i(G;\mathbf{Z}) = 0.</math> (This is equivalent to all homology groups other than ''H''<sub>0</sub> vanishing.)
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| ==Quasi-perfect group==
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| 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''<sup>(1)</sup> = ''G''<sup>(2)</sup> (the commutator of the commutator subgroup is the commutator subgroup), while a perfect group is one such that ''G''<sup>(1)</sup> = ''G'' (the commutator subgroup is the whole group). See {{harv|Karoubi|1973|pp=301–411}} and {{harv| Inassaridze | 1995 | p=76}}.
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| ==Notes==
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| {{reflist | group = note }}
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| ==References==
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| {{reflist}}
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| {{refbegin}}
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| * 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. {{MR|2009444}}
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| * {{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=Journal für Reine und Angewandte Mathematik | issn=0075-4102 | volume=174 | pages=1–14}}
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| *{{Citation | last1=Inassaridze | first1=Hvedri | title=Algebraic K-theory | url=http://books.google.com/?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}}
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| * Karoubi, M.: Périodicité de la K-théorie hermitienne, Hermitian K-Theory and Geometric Applications, Lecture Notes in Math. 343, Springer-Verlag, 1973
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| *{{Citation
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| | last = Rose
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| | first = John S.
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| | title = A Course in Group Theory
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| | publisher = Dover Publications, Inc.
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| | location = New York
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| | pages = 61
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| | year = 1994
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| | isbn = 0-486-68194-7
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| | mr = 1298629
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| }}
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| {{refend}}
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| ==External links==
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| * {{MathWorld|urlname=PerfectGroup|title=Perfect Group}}
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| * {{MathWorld|urlname=GruensLemma|title=Grün's lemma}}
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| [[Category:Group theory]]
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| [[Category:Properties of groups]]
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| [[Category:Lemmas]]
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