Show that every finite \(p\)-group has a nontrivial center.
Solution
Let \(|G|=p^n\). Under conjugation, the class equation is \[ |G| = |Z(G)|+ \sum_i [G:C_G(x_i)], \] where the \(x_i\) represent the noncentral conjugacy classes.
1. Every noncentral conjugacy-class size is divisible by \(p\).
For noncentral \(x_i\), the centralizer \(C_G(x_i)\) is a proper subgroup of the \(p\)-group \(G\). Thus its index \[ [G:C_G(x_i)] \] is a positive power of \(p\), hence divisible by \(p\).
2. Therefore \(p\mid |Z(G)|\).
Reduce the class equation modulo \(p\). Since \(p\mid|G|\) and every term in the sum is divisible by \(p\) by 1, \[ |Z(G)|\equiv0\pmod p. \]
3. Hence \(Z(G)\) is nontrivial.
The center contains the identity, so \(|Z(G)|\ge1\). By 2 its order is divisible by \(p\), hence \(|Z(G)|\ge p>1\).