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    $C_H(x)=H\cap C_G(x)$

    P-4H2GZ

    Area
    Algebra
    Topics
    Centralizers and Normalizers, Subgroups
    Status
    draft
    • Show that for \(H\leq G\), \(C_H(x) = H \intersect C_G(x)\).
    Solution

    By definition, \[ C_H(x)=\{h\in H:hx=xh\} \] and \[ C_G(x)=\{g\in G:gx=xg\}. \] Hence for \(h\in G\), \[ h\in C_H(x) \iff h\in H\text{ and }hx=xh \iff h\in H\cap C_G(x). \] Therefore \[ C_H(x)=H\cap C_G(x). \]

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