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Questions about the branch of algebra that deals with groups.

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Conjugate in the symmetric groups

Hi all, Let $p=2^n+1$ be an arbitrary (Fermat) prime. Consider $\pi=(1,2,...,p)\in S_p$, the symmetric group of degree $p$. My question is: Is there always a cycle, denoted by $\rho$, of length $p-1 …
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