Given a group action $f: G \times X \to X$, we say $f$ acts transitively on $X$ if for any $x \in X$, $O_x = X$ where $O_x$ denotes the Orbit of $x$
So, my question is as in the title:
If $C_{15}$ and $C_3$ are cyclic groups of orders $15$ and $3$, respectively, is it possible for a group action $f: C_{15} \times C_3 \to C_3$ to act transitively on $C_3$?
On the surface, I haven't been given any reason to see that this couldn't be the case. Clearly, the order of $C_3$ divides the order of $C_{15}$, so on this basis alone we don't have any reason to believe $f$ can't act transitively.
However, several attempts of mine to explicitly construct such a function $f$ have failed thus far, and after working on it for about an hour, I'm beginning to think it might not be possible. In particular, in every construction it has been difficult for me to get $f$ to be associative in the following sense: $f(gg', x) = f(g, f(g', x))$ for all $g, g' \in C_{15}$
Would there be any theoretical reason why $f$ couldn't act transitively, or if it is possible, would anyone be able to provide an example of such a group action?