Given an odd $n$, and an $m$ such that $(n,m)=1$, i would like to know what is the cycle structure of the permutation $\pi_{n,m} (a)=ma\bmod{n}$.
Specifically, how do i know if $\pi_{n,m}$ and $\pi_{n,k}$ have the same structure.
Even more specifically, do $\pi_{n,m}$ and $\pi_{n,m^{-1}}$ have the same structure, when $m\cdot{m^{-1}}=1\bmod{n}$.
Thanks!