Inner automorphism group

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.

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.