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Finite abelian group

  1. Mar 12, 2010 #1
    1. The problem statement, all variables and given/known data

    A be a finite abelian group, prove # of subgps of order p = # of subgps of index p, p is a prime.

    3. The attempt at a solution

    I have thought about this probably very easy problem for 2 hours and could not find a
    satisfying proof. I have tried bijective proof but failed
    (sending <x> |-> A/<x> is fruitless), and I tried elementary
    divisor decomposition
    (writing A as G x Z_(p^(a_1))x... x Z_(p^(a_n)) x H, focusing
    on the cyclic p-group part, and I have found that the number
    of order p subgroups must be p^n - 1 in any such A) which I think is the right
    direction but still can not work it out..T_T...please someone
    please give me some hint please...please don't refer to too advanced a
    theorem apart from the two abelian group decomposition theorems...thank you so much!!
    Last edited: Mar 12, 2010
  2. jcsd
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