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Math Help - finitely generated Abelian groups

  1. #1
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    finitely generated Abelian groups

    Identify all finitely generated Abelian groups of order 500 that have exactly 3 elements of order 2 and identify those elements.

    I think that finitely generated Abelian groups of order 500 are:

    \mathbb{Z}_2\oplus \mathbb{Z}_2\oplus\mathbb{Z}_5\oplus\mathbb{Z}_5\o  plus\mathbb{Z}_5\

     \mathbb{Z}_4\oplus\mathbb{Z}_5\oplus\mathbb{Z}_5\o  plus\mathbb{Z}_5\

    \mathbb{Z}_2\oplus \mathbb{Z}_2\oplus\mathbb{Z}_5\oplus\mathbb{Z}_{25  }\

    \mathbb{Z}_2\oplus \mathbb{Z}_2\oplus\mathbb{Z}_{125}

    \mathbb{Z}_4\oplus\mathbb{Z}_5\oplus\mathbb{Z}_{12  5}

    \mathbb{Z}_4\oplus \mathbb{Z}_{125}

    Is this correct?
    And if it is, which ones have exactly 3 elements of order 2?

    Any help or hint is appreciated.
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  2. #2
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    Quote Originally Posted by georgel View Post
    Identify all finitely generated Abelian groups of order 500 that have exactly 3 elements of order 2 and identify those elements.

    I think that finitely generated Abelian groups of order 500 are:

    \mathbb{Z}_2\oplus \mathbb{Z}_2\oplus\mathbb{Z}_5\oplus\mathbb{Z}_5\o  plus\mathbb{Z}_5\

     \mathbb{Z}_4\oplus\mathbb{Z}_5\oplus\mathbb{Z}_5\o  plus\mathbb{Z}_5\

    \mathbb{Z}_2\oplus \mathbb{Z}_2\oplus\mathbb{Z}_5\oplus\mathbb{Z}_{25  }\

    \mathbb{Z}_2\oplus \mathbb{Z}_2\oplus\mathbb{Z}_{125}

    \mathbb{Z}_4\oplus\mathbb{Z}_5\oplus\mathbb{Z}_{12  5} (wrong !)

    \mathbb{Z}_4\oplus \mathbb{Z}_{125}

    Is this correct?
    And if it is, which ones have exactly 3 elements of order 2?

    Any help or hint is appreciated.
    The below one seems like a typo.
    \mathbb{Z}_4\oplus\mathbb{Z}_5\oplus\mathbb{Z}_{12  5} should be changed to \mathbb{Z}_4\oplus\mathbb{Z}_5\oplus\mathbb{Z}_{25  }.
    Other than that, your classification looks OK to me.
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  3. #3
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    Quote Originally Posted by aliceinwonderland View Post
    The below one seems like a typo.
    \mathbb{Z}_4\oplus\mathbb{Z}_5\oplus\mathbb{Z}_{12  5} should be changed to \mathbb{Z}_4\oplus\mathbb{Z}_5\oplus\mathbb{Z}_{25  }.
    Other than that, your classification looks OK to me.
    Yes, it's a typo, thank you.

    Do you know how to determine which ones have 3 elements of order two? I know how to check whether element is of order two, but the idea is hardly to write down all elements and check..
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  4. #4
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    Quote Originally Posted by georgel View Post
    Yes, it's a typo, thank you.

    Do you know how to determine which ones have 3 elements of order two? I know how to check whether element is of order two, but the idea is hardly to write down all elements and check..
    In \mathbb{Z}_2 \oplus \mathbb{Z}_{250}, there are three elements of order 2, which are

    (1,0), (0, 125), (1, 125).

    Since \mathbb{Z}_2 \oplus \mathbb{Z}_{250} is isomorphic to \mathbb{Z}_2 \oplus \mathbb{Z}_{2} \oplus \mathbb{Z}_{125}, there are three elements of order 2 in \mathbb{Z}_2 \oplus \mathbb{Z}_{2} \oplus \mathbb{Z}_{125}.
    Last edited by aliceinwonderland; April 18th 2009 at 02:40 AM.
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  5. #5
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    Quote Originally Posted by aliceinwonderland View Post
    In \mathbb{Z}_2 \oplus \mathbb{Z}_{250}, there are three elements of order 2, which are

    (1,0), (0, 125), (1, 125).

    Since \mathbb{Z}_2 \oplus \mathbb{Z}_{250} is isomorphic to \mathbb{Z}_2 \oplus \mathbb{Z}_{2} \oplus \mathbb{Z}_{125}, there are three elements of order 2 in \mathbb{Z}_2 \oplus \mathbb{Z}_{2} \oplus \mathbb{Z}_{125}.
    Thank you again, I see that they are elements of order 2, I just fail to see how to get them myself. I'll think about it a bit longer.

    Thank you!
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