If $\displaystyle \phi:G\rightarrow K, \ \psi:H\rightarrow J$ are the isomorphisms, consider the map $\displaystyle \omega:G\oplus H\rightarrow K\oplus J$ defined by $\displaystyle \omega(g\oplus h)=\phi(g)\oplus\psi(h)$. Show this is an isomorphism.