Prime-to-p conjugacy class lemma ($Z(G)=1,p\mid |G|⟹\gcd (|\mathcal{C}|,p)=1\text{ for some nonidentity class}$)

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 $|G|\equiv 1(\mathrm{mod}p)$. Primality is essential: in the symmetric group $S_3$, neither nonidentity class size is coprime to the composite divisor $6$.