Automorphism group of a group

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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 $\operatorname{Aut}(G)$, 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 $\operatorname{Sym}(G)$, the group of all permutations on $G$.

Subgroups

Every group-closed automorphism property gives rise to a normal subgroup of the automorphism group. Some of the most important examples are given below:

Group-closed automorphism property Meaning Corresponding normal subgroup of the automorphism group
inner automorphism can be expressed as conjugation by an element of the group, i.e., there exists $g \in G$ such that the map has the form $x \mapsto gxg^{-1}$ it is called the inner automorphism group and is isomorphic to the quotient group $G/Z(G)$ where $Z(G)$ is the center. See group acts as automorphisms by conjugation.
class-preserving automorphism sends every element to within its automorphism class the class-preserving automorphism group
IA-automorphism sends every coset of the derived subgroup to itself, or equivalently, induces the identity map on the abelianization. the IA-automorphism group
center-fixing automorphism fixes every element of the center the center-fixing automorphism group
monomial automorphism can be expressed using a monomial formula the momomial automorphism group
normal automorphism sends every normal subgroup to itself the normal automorphism group

Facts

• Extensible equals inner: An automorphism of a group has the property that it can be extended to an automorphism for any bigger group containing it if and only if the automorphism is an inner automorphism.
• Quotient-pullbackable equals inner: An automorphism of a group has the property that it can be pulled back to an automorphism for any group admitting it as a quotient, if and only if the automorphism is an inner automorphism.