Inner automorphism

Definition

Let $G$ be a group and let $g \in G$. The inner automorphism given by $g$ is the map $c_g \colon G \to G$, $c_g(x) = gxg^{-1}$. The set of all inner automorphisms of $G$ is written $\operatorname{Inn}(G)$.

Remarks

Each $c_g$ is an automorphism of $G$, with inverse $c_{g^{-1}}$. Since $c_g \circ c_h = c_{gh}$, the map $g \mapsto c_g$ is a homomorphism from $G$ to $\operatorname{Aut}(G)$, and $\operatorname{Inn}(G)$ is its image. The subgroup $\operatorname{Inn}(G)$ is normal in $\operatorname{Aut}(G)$, because $\alpha \circ c_g \circ \alpha^{-1} = c_{\alpha(g)}$ for each automorphism $\alpha$. In an abelian group, each inner automorphism is the identity map.

Explanations

L1 human

Each element of a group gives a symmetry of the group: conjugate everything by this element. These symmetries are called inner, because they come from inside the group. In a group where all elements commute, conjugation changes nothing.