Inn(G) is a Normal Subgroup of Aut(G)
 Proposition 1: Let $G$ be a group. Then $\mathrm{Inn}(G)$ is a normal subgroup of $\mathrm{Aut}(G)$.
• Let $f \in \mathrm{Aut}(G)$ and let $i_a \in \mathrm{Inn}(G)$. Then for all $g \in G$ we have that:
• Thus $f \circ i_a \circ f^{-1} = i_{f(a)} \in \mathrm{Inn}(G)$ for each $f \in \mathrm{Aut}(G)$ and for each $i_a \in \mathrm{Inn}(G)$. So $f\mathrm{Inn}(G)f^{-1} \subseteq \mathrm{Inn}(G)$, i.e., $\mathrm{Inn}(G)$ is a normal subgroup of $\mathrm{Aut}(G)$.