Suppose we already know that for any groups $G$ and $H$, that if $f: G \rightarrow H$ is a homomorphism, then if $\langle g \rangle = G$ so too must $\langle f(g)\rangle = H$. Then we would know that if $f$ is a homomorphism, $f$ must "map generators to generators".
My question is to now flip this around and ask, "if we map generators to generators" then are we guaranteed that "$f$ is a homomorphism"?
So letting $G,H$ be groups suppose that $f: G \rightarrow H$ along with the stipulation that if $g \in G$ generates $G$ implies that $f(g)$ generates $H$.
Then we have for any $a,b \in G$ that $a = g^k$ and $b = g^j$ for some $k,j \in \mathbb{Z}$ so that
$ f(ab) = f(g^k g^j) = f(g^{j+k}) $
and from here it's not clear to me what we can conclude (other than that $f$ will be surjective). Specifically, I'm curious to know whether we can conclude $f$ is a homomorphism.