I am starting to self-study abstract algebra, and came across, in my humble opinion, a fairly difficult question. More specifically, the question I was trying to answer is along the lines of:
"Let $G$ a finite group, and $Hom(G,\Bbb C^\times)$ the set of all group homomorphisms $\phi:G\rightarrow\Bbb C^\times$, where $\Bbb C^\times$ is the multiplicative group of the non-zero complex numbers. Prove that $Hom(G,\Bbb C^\times)$ is finite."
I found many answers for the case where both groups were finite, but didn't have much success in this case.
I was trying to think about $G$ having a finite number of generators might lead to some brute-force technique, or maybe induction on the order of $G$, but no success up to now. I suspect there is something very basic I'm forgetting, so please go easy on me if that's the case.
I would be glad if you could give some insights about finding all group homomorphisms between groups too. What kind of facts are normally used to prove such a thing?
Thanks in advance.