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Math Help - Abelian Groups

  1. #1
    Super Member Aryth's Avatar
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    Abelian Groups

    Let G be an Abelian group. Let H and K be subgroups of G with orders 36 and 45, respectively. Which of the following is a possible order of HK?

    a) 5
    b) 9
    c) 20
    d) 90
    e) 180

    This doesn't seem difficult, but I've never taken Abstract Algebra before... So I don't even know what it means to find the order of a subgroup.
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  2. #2
    Member ModusPonens's Avatar
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    Re: Abelian Groups

    The order is the cardinal of the set in which you have the group structure. Have you talked about the 2nd isomorphism theorem? If yes, think how you can apply it to this problem.
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  3. #3
    Super Member Aryth's Avatar
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    Re: Abelian Groups

    I should have been clearer. I'm taking the course next semester, but I have a field exam that includes it this semester. I've never seen any of the isomorphism theorems, unfortunately.
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  4. #4
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    Re: Abelian Groups

    there is a standard result for this type of problem:

    |HK| = \frac{|H||K|}{|H \cap K|}

    for this problem, this means |HK| is a multiple of (36)(45)/gcd(36,45) = (36)(45)/9 = 180.

    thus (e) is the only correct answer.
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