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Normal Subgroup Conjugate of H by element

  1. Feb 9, 2012 #1

    Leb

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    1. The problem statement, all variables and given/known data
    Let H be a subgroup of group G. Then
    [itex]H \unlhd G \Leftrightarrow xHx^{-1}=H \forall x\in G [/itex]
    [itex] \Leftrightarrow xH=Hx \forall x\in G [/itex]
    [itex] \Leftrightarrow xHx^{-1}=Hxx^{-1} \forall x\in G [/itex]
    [itex] \Leftrightarrow xHx^{-1}=HxHx^{-1}=H \forall x\in G [/itex]
    2. Relevant equations
    xH={xh:h in H}

    3. The attempt at a solution
    Reviewed the definitions and properties of cosets/normal subgrps, but could not understand the last step. That is how to get from [itex] Hxx^{-1} to HxHx^{-1}[/itex]

    EDIT:
    Is it simply because [itex] xH=H \Leftrightarrow x \in H [/itex] ?
     
    Last edited: Feb 9, 2012
  2. jcsd
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