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Math Help - Finding Subgroups

  1. #1
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    Finding Subgroups

    Let G be the group (U(15), \times_{15}). Find all distinct subgroup of G with exactly two elements.

    Could anyone show me how to deal with this problem? So, I know that U(15)={1, 2, 4, 7, 8, 11, 13, 14} under multiplication modulo 15. I'm not sure if it helps, but I have constructed a Cayley table for G:


    Any help is very appreciated.
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  2. #2
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    Quote Originally Posted by demode View Post
    Let G be the group (U(15), \times_{15}). Find all distinct subgroup of G with exactly two elements.

    Could anyone show me how to deal with this problem? So, I know that U(15)={1, 2, 4, 7, 8, 11, 13, 14} under multiplication modulo 15. I'm not sure if it helps, but I have constructed a Cayley table for G:


    Any help is very appreciated.


    Of course it helps! A subgroup with two elements is of the form \{1,a\}\,,\,\,with\,\,\,a^2=1 , so locate all the elements with order 2 and
    form then your subgroups...

    Tonio
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  3. #3
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    And, of course, to do that you just have to look down the diagonal in your table.
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  4. #4
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    Okay then it's all the 1's in the main diagonal: g={1, 4, 11, 14}.

    And elements like 2 belong to order 4? Because 2 \times_{15} 2 = 4, 4 \times_{15} 2 = 8, 8 \times_{15} 2 = 1. Is that right?
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  5. #5
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    Quote Originally Posted by demode View Post
    Okay then it's all the 1's in the main diagonal: g={1, 4, 11, 14}.

    And elements like 2 belong to order 4? Because 2 \times_{15} 2 = 4, 4 \times_{15} 2 = 8, 8 \times_{15} 2 = 1. Is that right?

    Right indeed.

    Tonio
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