Let a finite group \(G\) act on itself by conjugation. The fixed points are the elements of the center \(Z(G)\), the orbit of \(x\) is its conjugacy class, and the stabilizer of \(x\) is its centralizer \(C_G(x)\). Consequently, \[ \abs G=\abs{Z(G)} +\sum_i [G:C_G(x_i)], \] where \(x_i\) runs over a set of representatives of the conjugacy classes of \(G\) not contained in \(Z(G)\).
(Finite $p$-groups have nontrivial center)
Let \(p\) be a prime and \(G\) a group with \(\abs G=p^a\), \(a>0\). Then \(p\) divides \(\abs{Z(G)}\), so \(Z(G)\neq\theset{e}\).
For \(x\notin Z(G)\), the conjugacy class of \(x\) has size \([G:C_G(x)]>1\), which divides \(p^a\) and is therefore a positive power of \(p\). Reducing the class equation modulo \(p\) gives \(\abs{Z(G)}\equiv\abs G\equiv0\pmod p\).