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Math Help - Subgroups of direct/semidirect products

  1. #1
    MHF Contributor Swlabr's Avatar
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    Subgroups of direct/semidirect products

    Let G=H \rtimes K (or even just G=H\times K).

    Is it true that if \bar{G}\leq G then either \bar{G}\leq H, \bar{G}\leq K, or \bar{G}=\bar{H}\rtimes \bar{K} where \bar{H}\leq H and \bar{K}\leq K?

    I know the equivalent isn't true for homomorphic images, but I'm not sure about semidirect products...
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  2. #2
    Member
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    Re: Subgroups of direct/semidirect products

    I don't think so.

    Let G=\mathbb{Z}_2\times\mathbb{Z}_2, and let \bar{G}=\langle(1,1)\rangle=\{(0,0),(1,1)\}. Certainly this subgroup is not contained in either factor of the direct product, and in this group, there isn't really much room left for it to be equal to the direct product of anything else.
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