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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). | |||
The | == Examples == | ||
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>). | |||
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. | |||
A non-trivial perfect group, however, is necessarily not [[solvable group|solvable]]. | |||
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. | |||
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,'' | |||
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 | |||
homology group ''H<sub>2</sub>(G)''. | |||
==Grün's lemma== | |||
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). | |||
<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): | |||
:<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|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> | |||
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''<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: | |||
* A [[superperfect group]] is one whose first two homology groups vanish: ''H''<sub>1</sub>(''G'', '''Z''') = ''H''<sub>2</sub>(''G'', '''Z''') = 0. | |||
* 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.) | |||
==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''<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}}. | |||
==Notes== | |||
{{reflist | group = note }} | |||
==References== | |||
{{reflist}} | |||
{{refbegin}} | |||
* 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}} | |||
* {{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}} | |||
*{{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}} | |||
* Karoubi, M.: Périodicité de la K-théorie hermitienne, Hermitian K-Theory and Geometric Applications, Lecture Notes in Math. 343, Springer-Verlag, 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:Group theory]] | |||
[[Category:Properties of groups]] | |||
[[Category:Lemmas]] | |||
Revision as of 20:34, 30 December 2013
In mathematics, more specifically in the area of modern algebra known as 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 (equivalently, its abelianization, which is the universal abelian quotient, is trivial). In symbols, a perfect group is one such that G(1) = G (the commutator subgroup equals the group), or equivalently one such that Gab = {1} (its abelianization is trivial).
Examples
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. 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 containing ).
More generally, a quasisimple group (a perfect 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 A5). 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 F2, 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.
Every acyclic group is perfect, but the converse is not true: A5 is perfect but not acyclic (in fact, not even superperfect), see Template:Harv. In fact, for n ≥ 5 the alternating group An is perfect but not superperfect, with H2(An, Z) = Z/2 for n ≥ 8.
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, 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 homology group H2(G).
Grün's lemma
A basic fact about perfect groups is Grün's lemma from Template:Harv: the quotient of a perfect group by its center is centerless (has trivial center).
Proof: If G is a perfect group, let Z1 and Z2 denote the first two terms of the upper central series of G (i.e., Z1 is the center of G, and Z2/Z1 is the center of G/Z1). If H and K are subgroups of G, denote the commutator of H and K by [H, K] and note that [Z1, G] = 1 and [Z2, G] ⊆ Z1, and consequently (the convention that [X, Y, Z] = [[X, Y], Z] is followed):
By the three subgroups lemma (or equivalently, by the Hall-Witt identity), it follows that [G, Z2] = [[G, G], Z2] = [G, G, Z2] = {1}. Therefore, Z2 ⊆ Z1 = Z(G), and the center of the quotient group G ⁄ Z(G) is the trivial group.
![{\displaystyle [Z_{2},G,G]=[[Z_{2},G],G]\subseteq [Z_{1},G]=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ffe55c439afb2112997df3ea9bf2fed853b6cb9)
![{\displaystyle [G,Z_{2},G]=[[G,Z_{2}],G]=[[Z_{2},G],G]\subseteq [Z_{1},G]=1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0018f0b71cd9fc08b926f4481c741707baf30039)
As a consequence, all 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: H1(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: H1(G, Z) = H2(G, Z) = 0.
- An acyclic group is one all of whose (reduced) homology groups vanish (This is equivalent to all homology groups other than H0 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 Template:Harv and Template:Harv.
Notes
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References
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro. Template:Refbegin
- 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. Template:MR
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15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - 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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Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
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External links
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