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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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    You are aware that this permutation can only be defined on the ring $\mathbb Z/n\mathbb Z$, otherwise it is not even close to being bijective.2011-05-22
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    yes, ofcourse... didn't even think of mentioning this...2011-05-22
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    It is a good practice in mathematics to ensure the reader knows the framework. Especially in such website of great variety in the topics.2011-05-22
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    IBS: (1) Your last edit makes Gerry's answer impossible to understand. (2) For general $m$ and $k$, $\pi_{n,m}$ and $\pi_{n,k}$ may have different cycle structures.2011-05-22
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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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