Difference between revisions of "Automorphism"
From Online Dictionary of Crystallography
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| − | ''f'' ('' | + | ''f'' (''g'') * ''f'' (''h'') = ''f'' (''g'' * ''h'') |
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| − | An automorphism | + | An automorphism preserves the structural properties of a group, e.g.: |
| + | * The identity element of ''G'' is mapped to itself. | ||
| + | * [[Subgroups]] are mapped to subgroups, [[normal subgroup]]s to normal subgroups. | ||
| + | * [[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 composition of two automorphisms is again an automorphism, and with | + | 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'' → ''G'' | + | ''f<sub>a</sub>'' : ''G'' → ''G'' |
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| − | ''f | + | ''f<sub>a</sub>(g)'' = ''aga<sup>−1</sup>'' |
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| − | + | for all ''g'' in ''G'', where ''a'' is a given fixed element of ''G''. | |
| − | The operation '' | + | The operation ''aga<sup>−1</sup>'' 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 [[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. | ||
==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'')'''. | + | 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 | + | For Abelian groups the mapping ''g'' → ''g<sup>-1</sup>'' is an outer automorphism, whereas for non-Abelian groups this mapping is not even a [[group homomorphism|homomorphism]]. |
==See also== | ==See also== | ||
Revision as of 10:39, 2 April 2009
Automorphisme (Fr). Automorphismus (Ge). Automorfismo (Sp). Automorfismo (It). 自己同形 (Ja).
Definition
An isomorphism from a group (G,*) to itself is called an automorphism of this group. It is a bijection f : G → G such that
f (g) * f (h) = f (g * h)
An automorphism preserves the structural properties of a group, e.g.:
- The identity element of G is mapped to itself.
- Subgroups are mapped to subgroups, normal subgroups to normal subgroups.
- Conjugacy classes are mapped to conjugacy classes (the same or another).
- 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 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 : G → G
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 center 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 g → g-1 is an outer automorphism, whereas for non-Abelian groups this mapping is not even a homomorphism.