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Thread: 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 $\displaystyle h\in K\cap N$, then,

    $\displaystyle h\in K$ and $\displaystyle h\in N$

    But $\displaystyle K,N$ are normal subgroups of $\displaystyle G$ ,so:

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

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

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

    Regards.

    Fernando Revilla
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