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Homework Help: Groups whose order is a power of a prime

  1. Sep 16, 2010 #1
    1. The problem statement, all variables and given/known data

    Does every group whose order is a power of a prime p contain an element of order p?

    2. Relevant equations



    3. The attempt at a solution
    I know it certainly can contain an element of order p. I also feel that
    |G|=|H|[G:H] might be useful. Any help is appreciated!
     
  2. jcsd
  3. Sep 16, 2010 #2
    Hmm, cool problem. I think the answer is yes, but I haven't thought it through completely, and more importantly I want to know your ideas. Have you tried using any of the corollaries of Lagrange's theorem that deal with the order (of a group or of an element in a group).
     
  4. Sep 16, 2010 #3

    Dick

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    It's called Cauchy's theorem.
     
  5. Sep 16, 2010 #4

    Office_Shredder

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    Cauchy's theorem is overkill for this.

    You can pick an arbitrary element of the group and construct an element of order p from it
     
  6. Sep 16, 2010 #5
    True. But we haven't got to the part about the center of a group yet.
     
  7. Sep 16, 2010 #6
    Yes I figured it out! That's what I did! Thanks!
     
  8. Sep 16, 2010 #7
    I figured it out. I just picked an element g of the group and proved that there is an element in <g> that has order p.
     
  9. Sep 16, 2010 #8
    Okay yeah I was slightly confused. At first I kept thinking that G had an order that was a multiple of p, and the tools I had did not solve it completely. When I realized that you actually stated ord(G) = p^n, I could do it (namely by using a^ord(G) = e and the fact that if a^m = e then ord(a)|m).

    Then Dick mentioned Cauchy's Theorem, so I briefly glanced at a proof on wikipedia and noticed that the tools used were more advanced. Finally I realized that Cauchy's Theorem is what I originally had in mind. So I guess the only thing left to do now is learn Cauchy's Theorem.
     
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