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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!

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    Did you try to solve your problem in the case $n$ is a prime? Then look at the case $n$ a prime power, and finally apply the Chinese Remainder Theorem.2011-05-23

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Those last two permutations are inverse to each other, no? What do you know about the cycle structure of a permutation and its inverse?

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    i'd still like an answer for the more general case, where $k$ and $m$ are ditinct, if there is a general criterion... thanks.2011-05-23