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Math Help - Def of normal subgroup

  1. #1
    Newbie jaco's Avatar
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    Def of normal subgroup

    The definition of a normal subgroup is: for each element, n, in N and each g in G, the element gng−1 is still in N where N is a subgroup of G. Can this be changed to ...element g−1ng is still in N. I think it still makes sense since gng−1=m for some m in N thus g−1gng−1g=g−1mg which gives n=g−1mg . Now g−1mg is in N since n is in N.
    Is this correct?
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  2. #2
    Super Member Gamma's Avatar
    Joined
    Dec 2008
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    Iowa City, IA
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    You sir are correct.

    Normally you write a subgroup N is normal in G iff gNg^{-1}\subset N for all g\in G. This is your definition given, but you can also notice that g\in G \Rightarrow g^{-1}\in G so this definition also tells you:
    (g^{-1})N(g^{-1})^{-1}=g^{-1}Ng\subset N

    Which is the second conclusion you drew.
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