Let $G$ and $G'$ be two finite groups such that $(\mathcal O(G),\mathcal O(G')) =1$. Then find all the homomorphisms from $G$ and $G'$. $\mathcal O(G)$ means order of group $G$. I tried this problem the whole day couldn't think of a way to solve. Really stuck.
Please help me out.
Hint:
If you denote $f:G\longrightarrow G'$ such a homomorphism, $f(G)$ is a subgroup of $G'$, hence its order divides that of $G'$. It is also isomorphic to a quotient of $G$ by the First isomorphism theorem and therefore its order divides that of $G$. Can you end the reasoning?