Past exam of the options II course of the University of Oxford / 2012 / B / B2 / f

Write $|G|=p^n$, where $n\ge 1$. By part (c), each conjugacy-class size divides $p^n$ and hence is a power of $p$. Every nonsingleton class consequently has size divisible by $p$. The class equation impliesThe identity element lies in the centre, so the centre is nonempty. Its size is a positive multiple of $p$, and thereforeThis proves that a nontrivial finite p-group has a nontrivial centre.