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Math Help - Group theory subnormality question

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    Group theory subnormality question

    How does one go about proving the subnormality intersection property, that is to say, if two subgroups are subnormal in G thier intersection is also?
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    Quote Originally Posted by prettyboy View Post
    How does one go about proving the subnormality intersection property, that is to say, if two subgroups are subnormal in G thier intersection is also?
    let H and K be two subnormal subgroups of G. let H=H_0 \lhd H_1 \lhd \cdots \lhd H_m=G and K=K_0 \lhd K_1 \lhd \cdots \lhd K_n=G be some subnormal series for H and K.

    then H \cap K=H_0 \cap K \lhd H_1 \cap K \lhd \cdots \lhd H_m \cap K=G \cap K=K \lhd K_1 \lhd \cdots \lhd K_n=G is a subnormal series for H \cap K.
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    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by prettyboy View Post
    How does one go about proving the subnormality intersection property, that is to say, if two subgroups are subnormal in G their intersection is also?
    Let H,K \unlhd G. Then for all h \in H \cap K and all g \in G we have that g^{-1}hg \in H and g^{-1}hg \in K, so clearly g^{-1}hg \in H \cap K, and so H \cap K \unlhd G holds.

    For more practice with the intersections of groups, can you prove that the intersection of finitely many subgroups of finite index is a subgroup of finite index?
    Last edited by Swlabr; June 8th 2009 at 10:58 PM.
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    Quote Originally Posted by Swlabr View Post
    Let H,K \unlhd G. Then for all h \in H \cap K and all g \in G we have that g^{-1}hg \in H and g^{-1}hg \in K, so clearly g^{-1}hg \in H \cap K, and so H \cap K \unlhd G holds.
    H and K are subnormal not normal. see the question again.
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    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by NonCommAlg View Post
    H and K are subnormal not normal. see the question again.

    Oooh-I did wonder what you were going on about in your post!

    That said, it is just a logical extension of the fact that I proved and so it's not an entirely irrelevant proof...
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