# Inner automorphism group

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## Definition

### Symbol-free definition

The **inner automorphism group** of a group is defined in the following equivalent ways:

- It is the subgroup of the automorphism group comprising those automorphisms that are inner, viz those automorphisms that arise as conjugation by an element.

- It is the quotient of the group by its center.

### Definition with symbols

The **inner automorphism group** of a group , denoted as , is defined in the following equivalent ways:

- It is the subgroup of comprising those automorphisms that are inner, viz those automorphisms that arise as conjugation by an element. That is, it is the set:

- It is the quotient where denotes the center of . In other words, it is the set of equivalence classes in under the relation of their
*ratio*being an element in the center.

### Equivalence of definitions

A group acts on itself as automorphisms by conjugation. Thus, there is a natural homomorphism that sends to the automorphism . The kernel of is the center of .

The inner automorphism group is the image of this homomorphism. Note that when we view it as a subgroup of , we get the first definition. When viewed as the quotient of by the kernel, we get the second definition.

## Facts

### Realization as inner automorphism group

Every group may not be realized as the inner automorphism group of some group. A group is termed a capable group if there is a group such that .

### Properties of the inner automorphism group

A group whose inner automorphism group is Abelian is termed a nilpotence class-2 group.

A group is nilpotent if and only if its inner automorphism group is nilpotent.