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Math Help - prove S_p is generated by a p-cycle and an arbitrary transposition

  1. #1
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    prove S_p is generated by a p-cycle and an arbitrary transposition

    Show that if p is prime, S_p=\langle\sigma,\tau\rangle where \sigma is any transposition and \tau is any p-cycle.
    Here S_p is the symmetric group of order p!. We must prove that an arbitrary transposition (that is, a cycle of length 2) and cycle of length p generate S_p.

    I can more or less guess intuitively how these work together, but it would be an enormous task to show it. I suspect there is an easier way, using homomorphisms or some such. Any hints would be much appreciated!
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  2. #2
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    Quote Originally Posted by hatsoff View Post
    Here S_p is the symmetric group of order p!. We must prove that an arbitrary transposition (that is, a cycle of length 2) and cycle of length p generate S_p.

    I can more or less guess intuitively how these work together, but it would be an enormous task to show it. I suspect there is an easier way, using homomorphisms or some such. Any hints would be much appreciated!

    I don't think there's an easy to do this: show that some power of that p-cycle times that transposition (or some already generated element) gives you all the transpositions and you're done.
    Try some particular cases> p = 3,5,7...

    Tonio
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