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| | They call me Emilia. My working day job is a librarian. California is our birth place. One of the issues she enjoys most is to read comics and she'll be beginning something else along with it.<br><br>Feel free to surf to my web blog - [http://www.redappledating.com/index.php?m=member_profile&p=profile&id=30210 http://www.redappledating.com/index.php?m=member_profile&p=profile&id=30210] |
| In [[abstract algebra]] an '''inner automorphism''' is a [[function (mathematics)|function]]
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| which, informally, involves a certain operation being applied, then another operation (shown as ''x'' below) being performed, and then the initial operation being reversed. Sometimes the initial action and its subsequent reversal change the overall result ("raise umbrella, walk through rain, lower umbrella" has a different result from just "walk through rain"), and sometimes they do not ("take off left glove, take off right glove, put on left glove" has the same effect as "take off right glove only").
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| More formally an inner automorphism of a [[group (mathematics)|group]] ''G'' is a function:
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| :ƒ: ''G'' → ''G''
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| defined for all ''x'' in ''G'' by
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| :ƒ(''x'') = ''a''<sup>−1</sup>''xa'',
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| where ''a'' is a given fixed element of ''G'', and where we deem the action of group elements to occur on the right (so this would read "a times x times a<sup>−1</sup>").
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| The operation ''a''<sup>−1</sup>''xa'' is called '''conjugation''' (see also [[conjugacy class]]), and it is often of interest to distinguish the cases where conjugation by one element leaves another element unchanged (as in the "gloves" analogy above) from cases where conjugation generates a new element (as in the "umbrella" analogy).
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| In fact, saying
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| :''a''<sup>−1</sup>''xa'' = ''x'' ("conjugation by ''a'' leaves ''x'' unchanged")
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| is equivalent to saying
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| :''ax'' = ''xa''. ("''a'' and ''x'' commute")
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| Therefore the existence and number of inner automorphisms that are not the [[identity mapping]] is a kind of measure of the failure of the [[commutative law]] in the group.
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| ==Notation==
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| The expression ''a''<sup>−1</sup>''xa'' is often denoted exponentially by ''x<sup>a</sup>''. This notation is used because we have the rule (''x<sup>a''</sup>)<sup>''b''</sup>=''x<sup>ab</sup>'' (giving a right [[group action|action]] of ''G'' on itself).
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| ==Properties==
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| Every inner automorphism is indeed an [[group automorphism|automorphism]] of the group ''G'', i.e. it is a [[bijective]] map from ''G'' to ''G'' and it is a [[group homomorphism|homomorphism]]; meaning (''xy'')<sup>''a''</sup> = ''x<sup>a</sup>y<sup>a</sup>''.
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| ==Inner and outer automorphism groups==
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| The [[functional composition|composition]] of two inner automorphisms is again an inner automorphism (as mentioned above: (''x<sup>a</sup>'')<sup>''b''</sup>=''x<sup>ab</sup>''), and with this operation, the collection of all inner automorphisms of ''G'' is itself a group, the inner automorphism group of ''G'' denoted Inn(''G'').
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| Inn(''G'') is a [[normal subgroup]] of the full [[automorphism group]] Aut(''G'') of ''G''. The [[quotient group]]
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| :Aut(''G'')/Inn(''G'')
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| is known as the [[outer automorphism group]] Out(''G''). The outer automorphism group measures, in a sense, how many automorphisms of ''G'' are not inner. Every non-inner automorphism yields a non-trivial element of Out(''G''), but different non-inner automorphisms may yield the same element of Out(''G'').
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| By associating the element ''a'' in ''G'' with the inner automorphism ƒ(''x'') = ''x<sup>a</sup>'' in Inn(''G'') as above, one obtains an [[group isomorphism|isomorphism]] between the [[quotient group]] ''G''/Z(''G'') (where Z(''G'') is the [[center of a group|center]] of ''G'') and the inner automorphism group:
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| :''G''/Z(''G'') = Inn(''G'').
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| This is a consequence of the [[isomorphism theorem|first isomorphism theorem]], because Z(''G'') is precisely the set of those elements of ''G'' that give the identity mapping as corresponding inner automorphism (conjugation changes nothing).
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| ===Non-inner automorphisms of finite ''p''-groups===
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| A result of Wolfgang Gaschütz says that if ''G'' is a finite non-abelian [[p-group|''p''-group]], then ''G'' has an automorphism of ''p''-power order which is not inner.
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| It is an open problem whether every non-abelian ''p''-group ''G'' has an automorphism of order ''p''.
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| The latter question has positive answer whenever ''G'' has one of the following conditions:
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| # ''G'' is nilpotent of class ''2''
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| # ''G'' is a [[Regular p-group|regular ''p''-group]]
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| # The centralizer C<sub>''G''</sub>(Z(<math>\Phi</math>(G))) in ''G'' of the center of the [[Frattini subgroup]] <math>\Phi</math>(G) of ''G'' is not equal to <math>\Phi</math>(G)
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| # ''G/Z(G)'' is a [[Powerful p-group|powerful ''p''-group]]
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| ===Types of groups===
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| It follows that the group Inn(''G'') of inner automorphisms is itself trivial (i.e. consists only of the [[identity element]]) [[if and only if]] ''G'' is [[abelian group|abelian]].
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| Inn(''G'') can only be a [[cyclic group]] when it is trivial, by a basic result on the center of a group.
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| At the opposite end of the spectrum, it is possible that the inner automorphisms exhaust the entire automorphism group; a group whose automorphisms are all inner and whose centre is trivial is called [[complete group|complete]]. This is the case for all of the symmetric groups on ''n'' elements when ''n'' is not 2 or 6: when ''n=6'' the symmetric group has a unique non-trivial class of outer automorphisms and when ''n=2'' the symmetric group is abelian, therefore its centre is non-trivial so that even though it has no outer automorphisms it nevertheless is not complete.
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| If the inner automorphism group of a [[perfect group]] ''G'' is simple, then ''G'' is called [[quasisimple group|quasisimple]].
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| ==Ring case==
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| Given a [[ring (mathematics)|ring]] ''R'' and a [[unit (ring theory)|unit]] ''u'' in ''R'', the map ƒ(''x'') = ''u''<sup>−1</sup>''xu'' is a [[ring homomorphism|ring automorphism]] of ''R''. The ring automorphisms of this form are called ''inner automorphisms'' of ''R''. They form a normal subgroup of the automorphism group of ''R''.
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| ==Lie algebra case==
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| An automorphism of a [[Lie algebra]] <math>\mathfrak{g}</math> is called an inner automorphism if it is of the form ''Ad''<sub>g</sub>, where ''Ad'' is the [[adjoint representation of a Lie group|adjoint map]] and ''g'' is an element of a [[Lie group]] whose Lie algebra is <math>\mathfrak{g}</math>. The notion of inner automorphism for Lie algebras is compatible with the notion for groups in the sense that an inner automorphism of a Lie group induces a unique inner automorphism of the corresponding Lie algebra.
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| ==Extension==
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| If ''G'' arises as the [[group of units]] of a [[ring theory|ring]] ''A'', then an inner automorphism on ''G'' can be extended to a mapping on the [[projective line over a ring|projective line over ''A'']] by the group of units of the [[matrix ring]] M<sub>2</sub>(''A''). In particular, the inner automorphisms of the [[classical group]]s can be so extended.
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| ==References==
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| * {{citation | mr=2574864 | last1=Abdollahi | first1=A. | title=Powerful ''p''-groups have non-inner automorphisms of order ''p'' and some cohomology | journal=J. Algebra | volume=323 | year=2010 | pages=779–789 }}
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| * {{citation | mr=2333188 | last1=Abdollahi | first1=A. | title=Finite ''p''-groups of class ''2'' have noninner automorphisms of order ''p'' | journal=J. Algebra | volume=312 | year=2007 | pages=876–879 }}
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| * {{citation | mr=1898386 | last1=Deaconescu | first1=M. | last2=Silberberg | first2=G. | title=Noninner automorphisms of order ''p'' of finite ''p''-groups | journal=J. Algebra | volume=250 | year=2002 | pages=283–287 }}
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| * {{citation | mr=0193144 | last1=Gaschütz | first1=W. | title=Nichtabelsche ''p''-Gruppen besitzen äussere ''p''-Automorphismen| journal=J. Algebra | volume=4 | year=1966 | pages=1–2 }}
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| * {{citation | mr=0173708 | last1=Liebeck | first1=H. | title=Outer automorphisms in nilpotent ''p''-groups of class ''2'' | journal=J. London Math. Soc. | volume=40 | year=1965 | pages=268–275 }}
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| *{{springer|title=Inner automorphism|id=I/i051230|last=Remeslennikov|first=V.N.}}
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| *{{MathWorld|title=Inner Automorphism|urlname=InnerAutomorphism}}
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| {{DEFAULTSORT:Inner Automorphism}}
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| [[Category:Group theory]]
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| [[Category:Group automorphisms]]
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| [[de:Automorphismus#Innere Automorphismen]]
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They call me Emilia. My working day job is a librarian. California is our birth place. One of the issues she enjoys most is to read comics and she'll be beginning something else along with it.
Feel free to surf to my web blog - http://www.redappledating.com/index.php?m=member_profile&p=profile&id=30210