# Automorphism

### From Online Dictionary of Crystallography

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* : *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

*f _{a}* :

*G*→

*G*

defined by

*f _{a}(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(**. The outer automorphism group is usually denoted by

*G*)**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.