= Solution
The <conjugacy class> of $x$ is its orbit under the <conjugation action>, and its stabilizer is the <centralizer> $C_G(x)$. The <orbit-stabilizer theorem> gives
$$
|\operatorname{Cl}_G(x)|=[G:C_G(x)].
$$
By <Lagrange's theorem>, the order of the <centralizer> divides $p^n=|G|$, so the index is a power of $p$. A <conjugacy class> has size one exactly when its element belongs to the <center of a group>.
Separate these singleton classes in the <class equation>:
$$
p^n=|Z(G)|+\sum_{x\notin Z(G)}[G:C_G(x)],
$$
where the sum takes one representative of each nonsingleton <conjugacy class>. Every summand is divisible by $p$. Thus $p\mid |Z(G)|$, and the identity ensures $|Z(G)|>0$. \b[Consequently $|Z(G)|\ge p$.] This is the <nontrivial center of a finite p-group> property.
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