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    $Z(G)\subseteq C_G(H)\subseteq N_G(H)$

    P-QEZAI

    Area
    Algebra
    Topics
    Centralizers and Normalizers, Subgroups
    Status
    draft
    • Show that \(Z(G) \subseteq C_G(H) \subseteq N_G(H)\).
    Solution

    If \(z\in Z(G)\), then \(zh=hz\) for every \(h\in H\), so \(z\in C_G(H)\). Hence \[ Z(G)\subseteq C_G(H). \] If \(g\in C_G(H)\), then for every \(h\in H\), \[ ghg^{-1}=h. \] Therefore \(gHg^{-1}=H\), so \(g\in N_G(H)\). Thus \[ Z(G)\subseteq C_G(H)\subseteq N_G(H). \]

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