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  1. #1
    Junior Member mathemanyak's Avatar
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    Help!

    (Let G be a group and let H ≤ G . If for all x ∈ G, (x^2)ϵH, then show that H is a normal subgroup of G,
    G/H is commutative quotient group.
    Please help
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  2. #2
    MHF Contributor

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    Quote Originally Posted by mathemanyak View Post
    (Let G be a group and let H ≤ G . If for all x ∈ G, (x^2)ϵH, then show that H is a normal subgroup of G,
    \forall x \in G, \ \forall h \in H: \ xhx^{-1}=h^{-1}(hx)^2(x^{-1})^2 \in H. \ \ \ \square

    G/H is commutative quotient group.
    since Hg=Hg^{-1}, \ \forall g \in G, for any x,y \in G we'll have: HxHy=Hxy=H(xy)^{-1}=Hy^{-1}x^{-1}=Hy^{-1}Hx^{-1}=HyHx. \ \ \square
    Last edited by NonCommAlg; July 22nd 2008 at 03:40 PM.
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