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Math Help - prove permutation is a cycle

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    prove permutation is a cycle

    Let p be a prime number. Prove that every power of a p-cycle is either a p-cycle or (1).
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    Quote Originally Posted by grad444 View Post
    Let p be a prime number. Prove that every power of a p-cycle is either a p-cycle or (1).
    Let \sigma be a p-cycle. Then you can form the subgroup (of the symmetric group) \left< \sigma\right> . Now \sigma^n \in \left< \sigma \right>. Since \left< \sigma \right> is a cyclic group of prime order it means the order of \sigma^n is either 1 or p. If it is 1 then \sigma^n = (1). Otherwise we know we can factor \sigma^n = \tau_1\cdot ... \tau_k where \tau_i's are disjoint cycles, if m_i is length of \tau_i then p=\text{ord}(\sigma^n) = \text{lcm}(m_1,...,m_k). But since m_i\not | p if k>1 (because it is prime) it means it is not possible for \text{lcm}(m_1,...,m_k) = p. Therefore k=1 and so \sigma^n must be a p-cycle.
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