In case the symbols aren't clear, $\displaystyle Z_{36} = \mathbb{Z}/36 \mathbb{Z}$, ie. residue classes mod 36. This one is written multiplicatively while $\displaystyle Z_{48}$ is written additively.Let $\displaystyle Z_{36} = <x>$. For which $\displaystyle a \in \mathbb{Z}$ does the map $\displaystyle \psi _a : Z_{48} \to Z_{36} : \overline{1} \mapsto x^a$ extend to a well defined homomorphism? Can $\displaystyle \psi _a$ ever be a surjective homomorphism?

I handled the a part of the question: all a such that (a, 36) = 1 will do the trick.

But isn't $\displaystyle \psi _a$ already a surjective homomorphism? Or am I supposed to show the modular nature of the problem creates the surjection?

Thanks!

-Dan