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Math Help - Number of subgroups of an abelian group

  1. #1
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    Number of subgroups of an abelian group

    How many subgroups of order p^{2} does the abelian group Z_{p^{3}} \oplus Z_{p^{2}} have?
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  2. #2
    Super Member Gamma's Avatar
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    Some things to consider

    A subgroup of an abelian group inherits commutativity, and we know the order of the subgroup is p^2, so we can apply the fundamental theorem of finitely generated groups to see that the only groups of order p^2 are \mathbb{Z}_{p^2} and \mathbb{Z}_{p}\oplus\mathbb{Z}_{p}. All that it amounts to is counting elements of order p and p^2 in the two groups, and then I think you can take it from there to count how many subgroups of this order you could make out of these elements.

    Hint: Theorem If d is a positive divisor of n, the number of elements of order d in a cyclic group of order n is \phi(d) where \phi is Euler's totient function. Euler's totient function - Wikipedia, the free encyclopedia it is the number of positive integers less than and relatively prime to n.
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