Can there be a homomorphism from $Z_4 ⊕Z_4$ onto $Z_8?$ Can there be a homomorphism from $Z_4 ⊕Z_4$ onto $Z_8?$
Attempt
Suppose $\psi :Z_4 ⊕Z_4 \rightarrow Z_8$ is a homomorphism
Then $|\ker \psi| = \dfrac {16}{8}=2$.
If $\psi (a,b) = c,$ then $O(c)$ divides $8$. 
Please tell me how do I proceed now? 
Thank you for your help!

If such a homomorphism exists, it must be that some $x\in \Bbb Z_4\oplus\Bbb Z_4$ maps to a generator in $\Bbb Z_8$. But the order of the image divides the order of the initial element, i.e. $o(\psi(a))\big|o(a)\big|4$, hence no image can have order $8$, and so no such homomorphism exists.
