Let H and K be groups with relatively prime orders. Show that Aut(HxK)≃Aut(H)xAut(K).
Given $\displaystyle \varphi \in \text{Aut}(H)$ and $\displaystyle \psi \in \text{Aut}(K)$, define a map $\displaystyle \text{Aut}(H) \times \text{Aut}(K) \to \text{Aut}(H\times K)$ by $\displaystyle f: (\varphi,\psi)(h,k)\mapsto (\varphi(h),\varphi(k))$. Show that this is a well defined homomorphism first, then show that it is bijective.