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Math Help - Normal Subgroups

  1. #1
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    Normal Subgroups

    Show that if K and N are normal subgroups of G, then K intersect N is normal to G.

    We have gkg^-1 is in K and gng^-1 is in N.
    I think I know the basic idea:
    Take a h in the intersection and take an arbitrary g.
    I need to show g^-1hg is in the intersection

    I guess I'm stuck in showing that. I guess maybe I don't understand what taking that h means.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by kathrynmath View Post
    I need to show g^-1hg is in the intersection
    Right. Consider h\in K\cap N, then,

    h\in K and h\in N

    But K,N are normal subgroups of G ,so:

    ghg^{-1}\in K and ghg^{-1}\in N for all g\in G .

    which implies ghg^{-1}\in K\cap N for all g\in G .

    As a consequence, K\cap N is a normal subgroup of G.

    Regards.

    Fernando Revilla
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