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Math Help - Normal Subgroups

  1. #1
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    Normal Subgroups

    I have to find the normal subgroups of D4xZ2 and I'm not too sure how to find them. I know that to be a normal subgroup the element has to commute with everything. So I know that I would have to check that (p1​, 1) commutes with everything, but I don't know how to check two elements like that at once.

    Any hints would be much appreciated.
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  2. #2
    MHF Contributor

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    Tejas
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    Re: Normal Subgroups

    no, to be a normal subgroup the right and left cosets have to be equal. obviously subgroups that lie in the center of D4xZ2 are normal, but there may be others.

    some things that may help you:

    any subgroup of order 8 will be normal.

    any subgroup that is in the center will be normal.

    the real trouble is going to be finding the subgroups of order 4 and 2. determining if a subgroup of order 2 is normal will be easy: <a> where a is an element of order 2 is normal if and only if <a> is in the center.
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