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 $G$, denoted as $Inn(G)$, is defined in the following equivalent ways:

• It is the subgroup of $Aut(G)$ comprising those automorphisms that are inner, viz those automorphisms that arise as conjugation by an element. That is, it is the set:

$\{\sigma \in Aut(G)|\exists g \in G , \sigma(x) = gxg^{-1} \forall x \in G \}$

• It is the quotient $G/Z(G)$ where $Z(G)$ denotes the center of $G$. In other words, it is the set of equivalence classes in $G$ 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 $c: G \to Aut(G)$ that sends $g$ to the automorphism $c_g = x \mapsto gxg^{-1}$. The kernel of $c$ is the center of $G$.

The inner automorphism group is the image of this homomorphism. Note that when we view it as a subgroup of $Aut(G)$, we get the first definition. When viewed as the quotient of $G$ 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 $H$ is termed a capable group if there is a group $G$ such that $H = G/Z(G)$.

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.