= Prime-to-p conjugacy class lemma
{title2=$Z(G)=1,\ p\mid|G|\ \Longrightarrow\ \gcd(|\mathcal C|,p)=1\text{ for some nonidentity class}$}
For a <finite group> with trivial <centre of a group> and a prime divisor $p$ of its order, some nonidentity <conjugacy class> has size <coprime> to $p$. Otherwise the <class equation> would give $|G|\equiv1\pmod p$. Primality is essential: in the <symmetric group> $S_3$, neither nonidentity class size is coprime to the composite divisor $6$.
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