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Contents 1 Definition 1.1 Symbol-free definition 1.2 Definition with symbols 2 Subgroups

This article is about a basic definition in group theory.The article text may, however, contain advanced material.
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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 , 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.