I am trying to determine for which integers $z$ the map $f_z : \mathbb{Z}/48\mathbb{Z} \rightarrow \Bbb{Z}_{36}$ with $f_z(\overline{1}) = x^a$, where $x$ is the generator of $\Bbb{Z}_{36}$, extends to homomorphism. I want to say that $f$ is a homomorphism only if it maps $\overline{1}$ to some generator in $\Bbb{Z}_{36}$. But that would require that something like the following is true:
Let $G$ and $H$ be cyclic groups of order $n$ and $m$, reps., with generators $g$ and $h$, reps. Then $f : G \rightarrow H$ is a homomorphism iff one of the generator in $G$ gets mapped to another in $H$.
Now, I imagine the claim as I presented it isn't true; it certainly isn't articulated very well. So, my question is, is there some theorem like the above that would prove useful for my problem; i.e., what other stipulations would we have to add?
EDIT:
Pmar seems to be claiming that $f_z$ need not map $\overline{1}$ to some generator, yet Bob Wilson's answer here, which has $3$ up-votes mind you, seems to suggest otherwise. Would someone mind addressing this ostensible discrepancy?