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Math Help - cauchy theorem

  1. #1
    ux0
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    cauchy theorem

    If p is a prime and G is a finite group in which every element has order a power of p, prove that G is a p-group.


    I was think cauchy theorem would help here.... If G is a finite group whose order is divisible by a prime p, the G contains an element of order p.
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  2. #2
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    Quote Originally Posted by ux0 View Post
    If p is a prime and G is a finite group in which every element has order a power of p, prove that G is a p-group.


    I was think cauchy theorem would help here.... If G is a finite group whose order is divisible by a prime p, the G contains an element of order p.

    Of course it helps...and a lot: if G is not a p-group then there's some prime q different from p that divides the order of G ==> by Cauchy's theorem G has an element of order q, contradiction.

    Tonio
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