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Math Help - Group and Direct Products

  1. #1
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    Group and Direct Products

    Let G be a group and let G=HxK. Give an example where J is a subgroup of G, but not of the form H_1xK_1 where H_1 is a subgroup of H and K_1 is a subgroup of K.
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  2. #2
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    Quote Originally Posted by pleasehelpme1 View Post
    Let G be a group and let G=HxK. Give an example where J is a subgroup of G, but not of the form H_1xK_1 where H_1 is a subgroup of H and K_1 is a subgroup of K.

    \{(0,0)\,,\,(1,1)\} \le \mathbb{Z}_2\times \mathbb{Z}_2

    Tonio
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  3. #3
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    ???

    what exactly do you mean by this? can you prove this for me? and what are the subgroup H_1 and K_1 in this case?
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by pleasehelpme1 View Post
    what exactly do you mean by this? can you prove this for me? and what are the subgroup H_1 and K_1 in this case?
    The only subgroups of \mathbb{Z}_2 are itself and the trivial subgroup....so what exactly can you conclude from tonio's example?
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  5. #5
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    ...

    That H_1 and K_1 must either be Z_2 or {0}. However, then J=Z_2xZ_2 or {0}x{0} contradicting the definition of J?
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  6. #6
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    ..

    and this is additive subgroups right?
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