Is Aut(G×H) isomorphic to Aut(G) × Aut(H)

Question

Please suggest me the proof.Am stuck with it.I saw somewhere that it will be true if (o(G),o(H))=1 but why?

Answer 1

Hint : Consider $H,K$ and the application $$Aut(H) \times Aut(K)\ni (\phi,\psi) \to ((h,k) \longmapsto (\phi(h),\psi(k))) \in Aut(H \times K)$$

Prove that $f$ (in general) is an injective homomorphism of group and is an isomorphism if and only if $H \times e_{K}$ and $e_{H} \times K$ are characteristic subgroups of $H \times K$.