1) I have to prove that if $\exists$ a nontrivial homomorphism $\phi:A\rightarrow B$, where A and B are finite and Abelian, then $|A|$ and $|B|$ are not relatively prime.
I know that $\phi(A)$ is a subgroup of $B$ and $\because$ $B$ is Abelian $\phi(A)\triangleleft B$. So I think $\phi(A) | |B|$ but I can't draw a connection between $|A|$ and $|B|$. Figured this out now. $A/Ker(\phi) \cong \phi(A)$, by Lagrange's, $|A/Ker(\phi)|$ divides $|A|$ and hence $|\phi(A)|$ divides $|A|$. Therefore, $|A|$ and $|B|$ are not relatively prime.
2) Deduce that $\exists$ a nontrivial homomorphism $B\rightarrow A$
Do we have to prove the converse of (1) because the orders are relatively prime? Figured it out. We can use the same argument as above.