# Automorphism group of a group

## Contents

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## Definition

### Symbol-free definition

The automorphism group of a group is defined as a group whose elements are all the automorphisms of the base group, and where the group operation is composition of automorphisms. In other words, it gets a group structure as a subgroup of the group of all permutations of the group.

### Definition with symbols

The automorphism group of a group $G$, denoted $Aut(G)$, is a set whose elements are automorphisms $\sigma:G \to G$, and where the group multiplication is composition of automorphisms. In other words, its group structure is obtained as a subgroup of $Sym(G)$, the group of all permutations on $G$.

## Subgroups

Every group-closed automorphism property gives rise to a normal subgroup of the automorphism group. Examples are the property of being an inner automorphism, class automorphism, extensible automorphism.