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Math Help - Normalizer of a Group

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    Normalizer of a Group

    Let G be a finite group and let H be a subgroup of G.
    NH, the normalizer of H in G is defined to be the set {x|xHx^-1}

    Prove that NH is a subgroup of G

    Prove that |{xHx^-1|x in G}| = |G/NH|
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  2. #2
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    Quote Originally Posted by Coda202 View Post
    Let G be a finite group and let H be a subgroup of G.
    NH, the normalizer of H in G is defined to be the set {x|xHx^-1}

    Prove that NH is a subgroup of G

    Prove that |{xHx^-1|x in G}| = |G/NH|
    Define the action of G on its subgroups by conjugation.
    Then the orbit of H is {xHx^{-1} | x in G} and the stabilizer of H is NH.
    Now it follows by orbit-stabilizer theorem that |{xHx^{-1}|x in G}| = |G/NH|.
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    Super Member Gamma's Avatar
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    already answered for you

    I already did this for you earlier in the week...
    http://www.mathhelpforum.com/math-he...t-algebra.html
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