  Let $G$ be a finite group and $a\in G$. Let $a^G=\{g^{-1}ag\mid g\in G\}$ be
the conjugacy class of $a$ in $G$. Assume that $a^G$ and $b^G$ are conjugacy
classes of $G$ with the property that ${\bf C}_G(a)={\bf C}_G(b)$. Then $a^G
b^G$ is a conjugacy class if and only if $[a,G]=[b,G]=[ab,G]$ and $[ab,G]$ is a
normal subgroup of $G$.
