Difference between revisions of "Automorphism"
From Online Dictionary of Crystallography
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==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 | + | 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 automorphism preserves the structural properties of a group, e.g. | + | 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. | ||
* [[Subgroup]]s are mapped to subgroups, [[normal subgroup]]s to normal subgroups. | * [[Subgroup]]s are mapped to subgroups, [[normal subgroup]]s to normal subgroups. | ||
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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. | + | 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== | ||
Revision as of 12:45, 12 May 2017
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 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 g → g-1 is an outer automorphism, whereas for non-Abelian groups this mapping is not even a homomorphism.