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    Every $p$-group has a nontrivial center

    E-AMD-D5ZZFZKC

    Area
    Algebra
    Topics
    Centralizers and Normalizers, Class Equation, p-Groups
    Status
    draft

    Show that every \(p\dash\)group has a nontrivial center.

    Solution

    Let \(G\) be a finite \(p\)-group. Its class equation is \[ |G|=|Z(G)|+\sum_i [G:C_G(x_i)], \] where the \(x_i\) represent the noncentral conjugacy classes.

    Each index \([G:C_G(x_i)]\) is a power of \(p\). Since \(x_i\) is noncentral, the index is greater than \(1\), hence divisible by \(p\). Also \(|G|\) is divisible by \(p\). Reducing the class equation modulo \(p\) gives \[ |Z(G)|\equiv0\pmod p. \] Since the identity belongs to \(Z(G)\), the center is nonempty, so its positive order is at least \(p\). Thus \[ Z(G)\ne1. \]

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