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Math Help - Centralizer of a Group

  1. #1
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    Centralizer of a Group

    Let G be a group. Let the centralixer C(X), X a subset of G, be defined as follows: C(X)={g in G | gx=xg for all x in X}.

    Prove the following:

    If X is a subset of Y, then C(Y) is a subset of C(X)

    X is a subset of C(C(X))

    C(X) = C(C(C(X)))
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  2. #2
    AMI
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    a) Suppose g\in C(Y). We want to show g\in C(X) i.e. gx=xg,\forall x\in X.
    Take x\in X\stackrel{X\subset Y}{\Longrightarrow}x\in Y\stackrel{g\in C(Y)}{\Longrightarrow}gx=xg.
    b) Suppose x\in X. We need x\in C(C(X))=\{g\in G | gy=yg,\forall y\in C(X)\}, so we want to prove that \forall y\in C(X) we have xy=yx.
    Take y\in C(X)\stackrel{x\in X}{\Longrightarrow}yx=xy.
    c)" \subset" Suppose g\in C(X). We want g\in C(C(C(X))) i.e. gy=yg,\forall y\in C(C(X)). Take y\in C(C(X))\stackrel{g\in C(X)}{\Longrightarrow}yg=gy.
    " \supset": From b) we know X\subset C(C(X))\stackrel{\text{a)}}{\Longrightarrow}C(C(C(  X)))\subset C(X).
    Last edited by AMI; June 12th 2009 at 12:21 PM.
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