I understand that by theorem, two groups are isomorphic, if and only if the two groups have the same elementary divisors. And the invariant factors can be simply found after that. So I'm trying to find elementary divisors and invariant factors of $U_{17}$ and $U_{45}$.
Since 17 is prime.. I highly doubt there will be a group such that will be isomorphic to $U_{17}$ but that's probably because I'm not very observant. Also, I need help in finding a group that would be isomorphic to $U_{45}$.
But I'm sure there could be an easier way that I'm not aware of. Please let me know if there is one.
Thank you.
Assuming that $U_n$ is the group of units of the ring $\mathbb Z/(n)$, we have:
$\quad U_{17} \cong C_{16}$
$\quad U_{45} \cong U_{9} \times U_{5} \cong C_{6} \times C_{4} \cong C_{2} \times C_{3} \times C_{4} \cong C_{2} \times C_{12} $.
The key facts are the Chinese remainder theorem and the fact that $U_{p^n}$ is cyclic for $p$ odd.