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Math Help - Normal subgroup

  1. #1
    Junior Member
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    Normal subgroup

    Let H be a group. Prove there is a group G containing a normal subgroup H' with the following two properties:

    (a) The group H is isomorphic to H' , and
    (b) For every automorphism  \phi of H', there is an element g\inG such that 
    \phi = I_{g} on H', where I_{g} is the inner automorphism associated to g.

    Thanks.
    Last edited by kierkegaard; October 3rd 2011 at 07:54 AM.
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  2. #2
    MHF Contributor Swlabr's Avatar
    Joined
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    Re: Normal subgroup

    Quote Originally Posted by kierkegaard View Post
    Let H be a group. Prove there is a group G containing a normal subgroup H' with the following two properties:

    (a) The group H is isomorphic to H' , and
    (b) For every automorphism \phi of H', there is an element g\inG such that
    \phi = I_{g} on H', where I_{g} is the inner automorphism associated to g.

    Thanks.
    Can you not just stack some semi-direct products up?...(so the induced automorphism is trivial on the extended bit, and induces the automorphism on H').

    EDIT: Alternatively, although this might be overkill, do you know what an HNN-extension is?...
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