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Difference between revisions of "Automorphism"

From Online Dictionary of Crystallography

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<Font color="blue">Automorphisme</font> (''Fr''). <Font color="red">Automorphismus</font> (''Ge''). <Font color="green">Automorfismo</font> (''Sp''). <Font color="black">Automorfismo</font> (''It''). <Font color="purple">自己同形</font> (''Ja'').
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<font color="blue">Automorphisme</font> (''Fr''). <font color="red">Automorphismus</font> (''Ge''). <font color="black">Automorfismo</font> (''It''). <font color="purple">自己同形</font> (''Ja''). <font color="green">Automorfismo</font> (''Sp'').
  
 
==Definition==
 
==Definition==
An [[Group isomorphism| isomorphism]] from a group (''G'',*) to itself is called an '''automorphism''' of this group. It is a bijection ''f'' : ''G'' → ''G'' such that
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An [[Group isomorphism|isomorphism]] from a group (''G'',*) to itself is called an '''automorphism''' of this group. It is a [[mapping|bijection]] ''f'' : ''G'' → ''G'' such that
  
 
<div align="center">
 
<div align="center">
''f'' (''u'') * ''f'' (''v'') = ''f'' (''u'' * ''v'')
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''f'' (''g'') * ''f'' (''h'') = ''f'' (''g'' * ''h'')
 
</div>
 
</div>
  
An automorphism always maps the identity to itself. The image under an automorphism of a [[conjugacy class]] is always a conjugacy class (the same or another). The [[image]] of an element has the same order as that element.
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An automorphism preserves the structural properties of a group, ''e.g.''
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* The identity element of ''G'' is mapped to itself.
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* [[Subgroup]]s are mapped to subgroups, [[normal subgroup]]s to normal subgroups.
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* [[Conjugacy class]]es are mapped to conjugacy classes (the same or another).  
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* The [[image]] ''f(g)'' of an element ''g'' has the same [[order]] as ''g''.
  
The composition of two automorphisms is again an automorphism, and with this operation the set of all automorphisms of a group ''G'', denoted by '''Aut(''G'')''', forms itself a group, the ''automorphism group'' of ''G''.
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The composition of two automorphisms is again an automorphism, and with composition as binary operation the set of all automorphisms of a group ''G'', denoted by '''Aut(''G'')''', forms itself a group, the ''automorphism group'' of ''G''.
  
 
==Inner automorphism==
 
==Inner automorphism==
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''f'' : ''G'' &rarr; ''G''  
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''f<sub>a</sub>'' : ''G'' &rarr; ''G''  
 
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</div>
  
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<div align="center">
''f''(''x'') = ''axa''<sup>−1</sup>
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''f<sub>a</sub>(g)'' = ''aga<sup>−1</sup>''
 
</div>
 
</div>
  
where ''a'' is a given fixed element of ''G'', for all ''x'' in ''G''.
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for all ''g'' in ''G'', where ''a'' is a given fixed element of ''G''.
  
The operation ''axa''<sup>−1</sup> is called '''conjugation''' (see also [[conjugacy class]]).  
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The operation ''aga<sup>−1</sup>'' is called '''conjugation''' by ''a'' (see also [[conjugacy class]]).
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The inner automorphisms form a [[normal subgroup]] of '''Aut(''G'')''', called the '''inner automorphism group''' and denoted by '''Inn(''G'')'''.
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The inner automorphism group is [[group isomorphism|isomorphic]] to the [[factor group|quotient]] of ''G'' by its [[center|centre]] ''Z(G)''. In particular, for [[Abelian group]]s the inner automorphism group consists just of the trivial automorphism.
  
 
==Outer automorphism==
 
==Outer automorphism==
The '''outer automorphism group''' of a group ''G'' is the [[factor group|quotient]] of the automorphism group '''Aut(''G'')''' by its inner automorphism group '''Inn(''G'')'''. The outer automorphism group is usually denoted '''Out(''G'')'''.
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The '''outer automorphism group''' of a group ''G'' is the [[factor group|quotient]] of the automorphism group '''Aut(''G'')''' by its inner automorphism group '''Inn(''G'')'''. The outer automorphism group is usually denoted by '''Out(''G'')'''.
  
For all Abelian groups there is at least the automorphism that replaces the group elements by their inverses. All other automorphisms are outer automorphisms. Non-Abelian groups have a non-trivial inner automorphism group, and possibly also outer automorphisms.
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For Abelian groups the mapping ''g'' &rarr; ''g<sup>-1</sup>'' is an outer automorphism, whereas for non-Abelian groups this mapping is not even a [[group homomorphism|homomorphism]].
  
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==See also==
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*[[Mapping]]
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*[[Group homomorphism]]
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*[[Group isomorphism]]
  
 
[[Category:Fundamental crystallography]]
 
[[Category:Fundamental crystallography]]

Latest revision as of 18:11, 8 November 2017

Automorphisme (Fr). Automorphismus (Ge). Automorfismo (It). 自己同形 (Ja). Automorfismo (Sp).

Definition

An isomorphism from a group (G,*) to itself is called an automorphism of this group. It is a bijection f : GG such that

f (g) * f (h) = f (g * h)

An automorphism preserves the structural properties of a group, e.g.

The composition of two automorphisms is again an automorphism, and with composition as binary operation the set of all automorphisms of a group G, denoted by Aut(G), forms itself a group, the automorphism group of G.

Inner automorphism

An inner automorphism of a group G is a function

fa : GG

defined by

fa(g) = aga−1

for all g in G, where a is a given fixed element of G.

The operation aga−1 is called conjugation by a (see also conjugacy class).

The inner automorphisms form a normal subgroup of Aut(G), called the inner automorphism group and denoted by Inn(G).

The inner automorphism group is isomorphic to the quotient of G by its centre Z(G). In particular, for Abelian groups the inner automorphism group consists just of the trivial automorphism.

Outer automorphism

The outer automorphism group of a group G is the quotient of the automorphism group Aut(G) by its inner automorphism group Inn(G). The outer automorphism group is usually denoted by Out(G).

For Abelian groups the mapping gg-1 is an outer automorphism, whereas for non-Abelian groups this mapping is not even a homomorphism.

See also