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| {{Group theory sidebar}}
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| In [[algebraic geometry]], an '''algebraic group''' (or '''group variety''') is a [[Group (mathematics)|group]] that is an [[algebraic variety]], such that the multiplication and inversion operations are given by [[regular function]]s on the variety.
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| In terms of [[category theory]], an algebraic group is a [[group object]] in the [[Category (mathematics)|category]] of algebraic varieties.
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| == Classes ==
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| Several important classes of groups are algebraic groups, including:
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| * [[Finite group]]s
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| * GL(''n'', '''C'''), the [[general linear group]] of [[invertible matrices]] over '''C'''
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| * [[Elliptic curve]]s.
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| Two important classes of algebraic groups arise, that for the most part are studied separately: ''[[abelian variety|abelian varieties]]'' (the 'projective' theory) and ''[[linear algebraic group]]s'' (the 'affine' theory). There are certainly examples that are neither one nor the other — these occur for example in the modern theory of [[differential of the first kind|integrals of the second and third kinds]] such as the [[Weierstrass zeta function]], or the theory of [[generalized Jacobian]]s. But according to [[Chevalley's structure theorem]] any algebraic group is an extension of an [[abelian variety]] by a linear algebraic group. This is a result of [[Claude Chevalley]]: if ''K'' is a [[perfect field]], and ''G'' an algebraic group over ''K'', there exists a unique normal closed subgroup ''H'' in ''G'', such that ''H'' is a linear group and ''G''/''H'' an abelian variety.
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| According to another basic theorem, any group in the category of [[affine variety|affine varieties]] has a [[Faithful representation|faithful]] finite dimensional [[linear representation]]: we can consider it to be a [[matrix group]] over ''K'', defined by polynomials over ''K'' and with matrix multiplication as the group operation. For that reason a concept of ''affine algebraic group'' is redundant over a field — we may as well use a very concrete definition. Note that this means that algebraic group is narrower than [[Lie group]], when working over the field of real numbers: there are examples such as the [[universal cover]] of the 2×2 special linear group that are Lie groups, but have no faithful linear representation. A more obvious difference between the two concepts arises because the [[identity component]] of an affine algebraic group ''G'' is necessarily of finite [[Index of a subgroup|index]] in ''G''.
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| When one wants to work over a base ring ''R'' (commutative), there is the [[group scheme]] concept: that is, a [[group object]] in the category of [[scheme (mathematics)|scheme]]s over ''R''. ''Affine group scheme'' is the concept dual to a type of [[Hopf algebra]]. There is quite a refined theory of group schemes, that enters for example in the contemporary theory of abelian varieties.
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| ==Algebraic subgroup==
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| An '''algebraic subgroup''' of an algebraic group is a [[Zariski topology|Zariski closed]] [[subgroup]].
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| Generally these are taken to be connected (or irreducible as a variety) as well.
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| Another way of expressing the condition is as a [[subgroup]] which is also a [[algebraic variety|subvariety]].
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| This may also be generalized by allowing [[scheme (mathematics)|schemes]] in place of varieties. The main effect of this in practice, apart from allowing subgroups in which the [[connected space|connected component]] is of finite index > 1, is to admit non-[[reduced scheme]]s, in characteristic ''p''.
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| ==Coxeter groups==
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| {{main|Coxeter group}}
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| {{further|Field with one element}}
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| There are a number of analogous results between algebraic groups and [[Coxeter group]]s – for instance, the number of elements of the symmetric group is <math>n!</math>, and the number of elements of the general linear group over a finite field is the [[q-factorial|''q''-factorial]] <math>[n]_q!</math>; thus the symmetric group behaves as though it were a linear group over "the field with one element". This is formalized by the [[field with one element]], which considers Coxeter groups to be simple algebraic groups over the field with one element.
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| ==See also==
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| *[[Algebraic topology (object)]]
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| *[[Borel subgroup]]
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| *[[Tame group]]
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| *[[Morley rank]]
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| *[[Cherlin–Zilber conjecture]]
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| *[[Adelic algebraic group]]
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| *[[Glossary of algebraic groups]]
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| ==Notes==
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| <references/>
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| == References ==
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| *{{Citation | editor1-last=Chevalley | editor1-first=Claude | title=Séminaire C. Chevalley, 1956--1958. Classification des groupes de Lie algébriques | url=http://www.numdam.org/numdam-bin/browse?id=SCC_1956-1958__1_ | publisher=Secrétariat Mathématique | location=Paris | series=2 vols | mr=0106966 |id= Reprinted as volume 3 of Chevalley's collected works. | year=1958}}
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| * {{Citation | last1=Humphreys | first1=James E. | title=Linear Algebraic Groups | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Graduate Texts in Mathematics | isbn=978-0-387-90108-4 | mr=0396773 | year=1972 | volume=21}}
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| * {{Citation | last1=Lang | first1=Serge | author1-link=Serge Lang | title=Abelian varieties | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-90875-5 | year=1983}}
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| * Milne, J. S., ''[http://www.jmilne.org/math/CourseNotes/ala.html Affine Group Schemes; Lie Algebras; Lie Groups; Reductive Groups; Arithmetic Subgroups]''
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| * {{Citation | last1=Mumford | first1=David | author1-link=David Mumford | title=Abelian varieties | publisher=[[Oxford University Press]] | isbn=978-0-19-560528-0 | oclc=138290 | year=1970}}
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| * {{Citation | last1=Springer | first1=Tonny A. | title=Linear algebraic groups | publisher=Birkhäuser Boston | location=Boston, MA | edition=2nd | series=Progress in Mathematics | isbn=978-0-8176-4021-7 | mr=1642713 | year=1998 | volume=9}}
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| * {{Citation | last1=Waterhouse | first1=William C. | authorlink = William C. Waterhouse | title=Introduction to affine group schemes | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Graduate Texts in Mathematics | isbn=978-0-387-90421-4 | year=1979 | volume=66}}
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| * {{Citation | last1=Weil | first1=André | author1-link=André Weil | title=Courbes algébriques et variétés abéliennes | publisher=Hermann | location=Paris | oclc=322901 | year=1971}}
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| [[Category:Algebraic groups| ]]
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| [[Category:Properties of groups]]
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| [[Category:Algebraic geometry]]
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| [[Category:Algebraic varieties]]
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52 yrs old Geophysicist Les from Rockland, has many hobbies that include snowshoeing, world at arms hack and dolls. Was exceptionally stimulated after visiting Bisotun.
My blog post :: world at arms ios Hack no survey