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Difference between revisions of "Automorphism"

From Online Dictionary of Crystallography

(adjusted inner automorphisms, some further small changes)
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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 [[mapping|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
  
 
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</div>
 
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An automorphism preserves the structural properties of a group, e.g.:
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An automorphism preserves the structural properties of a group, ''e.g.''
 
* The identity element of ''G'' is mapped to itself.
 
* The identity element of ''G'' is mapped to itself.
* [[Subgroups]] are mapped to subgroups, [[normal subgroup]]s to normal subgroups.
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* [[Subgroup]]s are mapped to subgroups, [[normal subgroup]]s to normal subgroups.
 
* [[Conjugacy class]]es are mapped to conjugacy classes (the same or another).  
 
* [[Conjugacy class]]es are mapped to conjugacy classes (the same or another).  
 
* The [[image]] ''f(g)'' of an element ''g'' has the same [[order]] as ''g''.
 
* The [[image]] ''f(g)'' of an element ''g'' has the same [[order]] as ''g''.
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The inner automorphisms form a [[normal subgroup]] of '''Aut(''G'')''', called the '''inner automorphism group''' and denoted by '''Inn(''G'')'''.
 
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 [[group isomorphism|isomorphic]] to the [[factor group|quotient]] of ''G'' by its [[center]] ''Z(G)''. In particular, for [[Abelian group]]s the inner automorphism group consists just of the trivial automorphism.
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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==

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