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Math Help - Normalizer of a cyclic group

  1. #1
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    Normalizer of a cyclic group

    Let G be a finite group and let x be in G.

    Prove that if g\in N_G(<x>), then gxg^{-1}=x^a for some a\in\mathbb{Z}.

    N_G(<x>)=\{g\in G:g<x>g^{-1}=x\}

    Let |x|=n

    <x>=\{e,x,x^2,\cdots x^{n-1}\}

    Now what?
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  2. #2
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    Re: Normalizer of a cyclic group

    Quote Originally Posted by dwsmith View Post
    Let G be a finite group and let x be in G.

    Prove that if g\in N_G(<x>), then gxg^{-1}=x^a for some a\in\mathbb{Z}.

    N_G(<x>)=\{g\in G:g<x>g^{-1}=x\}

    Let |x|=n

    <x>=\{e,x,x^2,\cdots x^{n-1}\}

    Now what?
    Note that N_G(<x>)=\{g\in G:g<x>g^{-1}=<x>\}.

    Now, x\in<x> \implies gxg^{-1}\in g<x>g^{-1} \overset{g\in N_G(<x>)}{==}<x>.

    So, gxg^{-1}=x^a for some a\in\mathbb{Z}.
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