Let \(p\) be any prime number and let \(n \ge 1\) be an integer.
Let \(G\) be a group with order \(p^n\). We show that \(Z(G)\) is not trivial.
Suppose not. That is, \(Z(G) = (1)\).
Consider the class equation of \(G\). The size of any nontrivial orbit will be of the form \(p^{m}\) for some \(1 \le m \le n\). (By orbit-stabiliser theorem.) Thus, we get:
\(|G| = p^n = 1 + p^{a_1} + \cdots + p^{a_k}\)
However, this is a contradiction as the right hand side of the equation is not divisible by \(p.\) QED.