Automorphism group ( $Aut (G)$ )

The automorphism group of a group

G

is the group of all isomorphisms

G \to G

under composition.

Table of contents
- Inner automorphism Automorphism group
- Outer automorphism group Automorphism group

Inner automorphism ( $Inn (G)$ )

An inner automorphism is conjugation by an element of the group. The inner automorphisms form a normal subgroup

Inn (G) ◃ Aut (G)

.

Outer automorphism group ( $Out (G)$ )

The outer automorphism group is

Out (G) = Aut (G) / Inn (G) .

It records automorphisms modulo conjugation by elements of

G

.

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Automorphism group by

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In mathematics, specifically in the field of group theory and abstract algebra, an automorphism group is a concept that involves the symmetries of a mathematical structure. ### Definition An **automorphism** is an isomorphism from a mathematical object to itself. In other words, it is a bijective mapping that preserves the structure of the object.

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