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P-group homework question

  1. Feb 24, 2008 #1
    1. The problem statement, all variables and given/known data

    If G is a p group, show that it has a subgroup of order p^m for every 0<=m<=n.

    3. The attempt at a solution

    The only thing I know about p-groups is that they have nontrivial centers.
  2. jcsd
  3. Feb 24, 2008 #2


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    I take it that the order of G is p^n? And I take it you're not supposed to use Sylow?

    How about induction on n then?
  4. Feb 24, 2008 #3
    yes, p-group is by definition a group of order p^n. And it's OK to use Sylow theorems, but how?
  5. Feb 24, 2008 #4
    Well, it has another definition where every element has order p^k for some k. Anyway, I think induction is pretty much the only way to go. As I recall, this proof is rather tricky so don't get discouraged.
  6. Feb 24, 2008 #5


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    This is really a restatement of one of the Sylow theorems for p-groups. So it's a good idea to try to study a proof of the appropriate Sylow theorem.
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