Prove there is no homomorphism from $Z_{16} \oplus Z_2 $ ont $Z_4 \oplus Z_4$

Question

Prove there is no homomorphism from $Z_{16} \oplus Z_2 $ onto $Z_4 \oplus Z_4$

I have no idea at all how to attempt such question. I have read some solutions in which we find the cardinality of $kerm\phi$ for some $\phi mapping $ that is onto using the first theorem of homomorphism.

After that I don't understand anything

Is there a general approach to these questions ?

Answer 1

If there were a homomorphism, the kernel would be a subgroup of order $2$, since $Z_4 \oplus Z_4$ has order $16$, and the first isomorphism theorem implies that $Z_{16} \oplus Z_2/ker \phi \cong Z_4 \oplus Z_4$ There is only one subgroup of order $2$ in this group, and its quotient is not isomorphic to $Z_4 \oplus Z_4$.

Answer 2

Write $a$ and $b$ for the generators of the $Z_{16}$ and $Z_2$ on the left. If $\phi$ is a homomorphism from $Z_{16}\oplus Z_2$ to $Z_{4}\oplus Z_4$ then $\phi(4a)=0$. This means that the kernel of $\phi$ has size at least $4$. Can $\phi$ be onto with a kernel that big?