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Math Help - normal subgroup

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    normal subgroup

    Assume H and K are normal subgroups of G with the intersection of H and K is equal to 1. Prove xy = yx for every x in H and y in K. (That is, to show x^{-1}y^{-1}xy is in the intersection of H and K)
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    Quote Originally Posted by dori1123 View Post
    Assume H and K are normal subgroups of G with the intersection of H and K is equal to 1. Prove xy = yx for every x in H and y in K. (That is, to show x^{-1}y^{-1}xy is in the intersection of H and K)
    x(yx^{-1}y^{-1}) \in H \text{ and }(xyx^{-1})y^{-1} \in K.
    Therefore, xyx^{-1}y^{-1} = e and so xy = yx.
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    Quote Originally Posted by ThePerfectHacker View Post
    x(yx^{-1}y^{-1}) \in H \text{ and }(xyx^{-1})y^{-1} \in K.
    How do you know x(yx^{-1}y^{-1}) \in H ?
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  4. #4
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    Quote Originally Posted by dori1123 View Post
    How do you know x(yx^{-1}y^{-1}) \in H ?
    Because x \in H.
    Because yxy^{-1} \in H -- this is by normality.
    Therefore x(yxy^{-1}) \in H by closure on H.
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