Show that \(N_G(H) / C_G(H)\) is isomorphic to a subgroup of \(\operatorname{Aut}(H)\).
Solution
Conjugation by an element of \(N_G(H)\) preserves \(H\), so there is a homomorphism \[ \Phi:N_G(H)\to\operatorname{Aut}(H),\qquad \Phi(g)(h)=ghg^{-1}. \] Its kernel consists exactly of those \(g\) commuting with every element of \(H\), namely \[ \ker\Phi=C_G(H). \] By the first isomorphism theorem, \[ N_G(H)/C_G(H)\cong\operatorname{im}\Phi\le\operatorname{Aut}(H). \]