Inner Automorphism

Let $G$ be a group . For each $g\in G$, the map

is an automorphism, called the inner automorphism determined by $g$. The set of all inner automorphisms is a subgroup $\operatorname{Inn}(G)\le \operatorname{Aut}(G)$ of the automorphism group .

The assignment $g\mapsto c_g$ is a homomorphism $G\to \operatorname{Inn}(G)$ whose kernel is the center . Hence there is a natural isomorphism

a quotient group measuring how far $G$ is from being abelian.

Examples:

• If $G$ is abelian, then $c_g=\mathrm{id}_G$ for all $g$, so $\operatorname{Inn}(G)$ is trivial.
• In $S_3$, the center is trivial, so $\operatorname{Inn}(S_3)\cong S_3$.
• Inner automorphisms are exactly the permutations of $G$ arising from the conjugation action .