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Math Help - Product of two prime cycles

  1. #1
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    Product of two prime cycles

    Let \sigma,\tau\in{S_p} be cycles of length p, where p is a prime. Prove or disprove that \sigma\tau is a cycle of length p.
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  2. #2
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    Quote Originally Posted by bulls6x View Post

    Let \sigma,\tau\in{S_p} be cycles of length p, where p is a prime. Prove or disprove that \sigma\tau is a cycle of length p.
    it's trivially false: for example if \tau=\sigma^{-1}, you'll get a counter-example. a better question is to see if we can find all cycles \sigma, \tau \in S_n of length n such that \sigma \tau is also a cycle of length n.

    another question (probably easier): is it always true that if \sigma, \tau and \sigma \tau are cycles of length n in S_n, then \tau \sigma is also a cycle of length n?
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    Quote Originally Posted by bulls6x View Post
    Let \sigma,\tau\in{S_p} be cycles of length p, where p is a prime. Prove or disprove that \sigma\tau is a cycle of length p.
    Well if \tau = \sigma^{-1} then  \sigma \tau = \text{ id}. Thus, I guess you want to say \sigma\tau is a p-cycle or the identity. However, that still doth not work because (12345)(15234) = (24)(35).

    EDIT: NonCommAlg responded faster.
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