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Math Help - Centralizer of Group

  1. #1
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    Centralizer of Group

    Let G be a finite group with g in G.
    Ng, the centralizer of g in G is defined to be the set = {x|xgx^-1 = g}

    Show that Ng is a subgroup of G.

    Now, letting Sg = {xgx^-1|x in G}

    Prove that, |Sg| = |G/Ng|
    and that {Sg|g in G} is a partition of G
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  2. #2
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    Quote Originally Posted by Coda202 View Post
    Show that Ng is a subgroup of G.
    Just check the definitions of being a subgroup.

    Prove that, |Sg| = |G/Ng|
    and that {Sg|g in G} is a partition of G
    Hint: aga^{-1} = bgb^{-1} iff (b^{-1}a)g(b^{-1}a)^{-1} = g iff a(Ng) = b(Ng).
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