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Math Help - isomorphic to product

  1. #1
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    isomorphic to product

    Hi, if someone can explain how to get the answer to this is would be greatly appreciated.

    Let:
    G be the set of matrices of the form:
    [a b : 0 1]
    H be the set of matrices of the form:
    [a 0 : 0 1]
    K be the set of matrices of the form:
    [1 b : 0 1]

    Q1: Is G isomorphic to H x K? Prove it, or give a counterexample.
    Q2: Is G/K isomorphic to H? Prove it, or give a counterexample.

    Any help would be nice.

    Thanks
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Berkeley, California
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    Re: isomorphic to product

    Quote Originally Posted by shelford View Post
    Hi, if someone can explain how to get the answer to this is would be greatly appreciated.

    Let:
    G be the set of matrices of the form:
    [a b : 0 1]
    H be the set of matrices of the form:
    [a 0 : 0 1]
    K be the set of matrices of the form:
    [1 b : 0 1]

    Q1: Is G isomorphic to H x K? Prove it, or give a counterexample.
    Q2: Is G/K isomorphic to H? Prove it, or give a counterexample.

    Any help would be nice.

    Thanks
    What have you tried? Please show some work.
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  3. #3
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    Joined
    Jul 2011
    Posts
    21

    Re: isomorphic to product

    For 1: to show G is isomorphic to H x K I need to show:
    i) G=HK
    ii) H \cap K = I
    ii) H, K are normal subgroups of G

    for i) HK=[a ab : 0 1]
    ii) H \cap K = \{ 1 \}

    Not sure if this is correct and where to go now.
    How do I prove that H and K are normal subgroups of G?

    And I don't know how to do q2 at all.

    Thanks
    Last edited by shelford; September 6th 2011 at 07:44 PM.
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