= Solution
Write $|G|=p^n$, where $n\geq1$. 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> implies
$$
|Z(G)|\equiv|G|\equiv0\pmod p.
$$
The <identity element> lies in the <centre>, so the <centre> is nonempty. Its size is a positive multiple of $p$, and therefore
$$
\boxed{|Z(G)|\geq p\geq2.}
$$
This proves that a nontrivial finite <p-group> has a nontrivial <centre>.
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