Show that any automorphism (and in particular, every inner automorphism) sends conjugate subgroups to conjugate subgroups.
Solution
Suppose \(H_2=gH_1g^{-1}\) and let \(\varphi\in\operatorname{Aut}(G)\). Then \[ \varphi(H_2) =\varphi(gH_1g^{-1}) =\varphi(g)\,\varphi(H_1)\,\varphi(g)^{-1}. \] Thus \(\varphi(H_1)\) and \(\varphi(H_2)\) are conjugate subgroups of \(G\). In particular, this holds for every inner automorphism.