Inner automorphism

This article is about a basic definition in group theory. The article text may, however, contain advanced material.
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This article defines an automorphism property, viz a property of group automorphisms. Hence, it also defines a function property (property of functions from a group to itself)
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Definition

Symbol-free definition

An automorphism of a group is termed an inner automorphism if it can be expressed as conjugation by an element of the group.

Note that the choice of conjugating element is not unique, in fact the possibilities for the conjugating element form a coset of the center.

Definition with symbols

An automorphism $\sigma$ of a group $G$ is termed an inner automorphism if there is an element $g$ in $G$ such that for all $x$, $\sigma (x)=gxg^{-1}$.

Justification for the definition

The notion of inner automorphism makes good sense because of the following fact: a group acts on itself as automorphisms via the conjugation map. This has the following consequences:

• Every conjugation actually defines an automorphism
• There is a homomorphism from the group to its automorphism group that sends each element to the corresponding conjugation map.

Facts

Homomorphism from the group to its automorphism group

The kernel of the natural homomorphism from a group to its automorphism group is the center of the group. This is because the condition that conjugation by an element be the identity map is equivalent to the condition that it commute with every element. The center of a group $G$ is denoted as $Z(G)$. The image, which is the inner automorphism group, is thus $G/Z(G)$.

Equivalence relation on elements

Two elements in a group are termed conjugate if they are in the same orbit under the action of the group by conjugation.

Metaproperties

Group-closedness

This automorphism property is group-closed: it is closed under the group operations on automorphisms (composition, inversion and the identity map). It follows that the subgroup comprising automorphisms with this property, is a normal subgroup of the automorphism group
View a complete list of group-closed automorphism properties

A composite of inner automorphisms is inner, and an inverse of an inner automorphism is inner. The identity map is clearly inner. Hence, the inner automorphisms form a subgroup of the automorphism group, termed the inner automorphism group