Automorphism group of a group

This article is about a basic definition in group theory. The article text may, however, contain advanced material.
VIEW: Definitions built on this | Facts about this: (facts closely related to Automorphism group of a group, all facts related to Automorphism group of a group) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |[SHOW MORE]

View a complete list of basic definitions in group theory | Go through a guided tour for beginners to this wiki

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.