Let $G$ and $H$ be groups, $a,b\in G$ and $f: G\to H$ be a homomorphism. We were tasked to show that
b) if $f(a)$ has finite order in $H$, then $|a|$ is either infinite or $|f(a)|$ divides $|a|.$
I assume b could be answered when I have showed that a holds, but what I only have in my scratch solution is that since $f$ is a homomorphism then $f(ab)=f(a)f(b)$. I'm stuck. Just a hint on how to continue would be much appreciated.
Thank you very much.