ON $p$ -PARTS OF CONJUGACY CLASS SIZES OF FINITE GROUPS

Let $G$ be a finite group. Let $\operatorname{cl}(G)$ be the set of conjugacy classes of $G$ and let $\operatorname{ecl}_{p}(G)$ be the largest integer such that $p^{\operatorname{ecl}_{p}(G)}$ divides $|C|$ for some $C\in \operatorname{cl}(G)$ . We prove the following results. If $\operatorname{ecl}_{p}(G)=1$ , then $|G:F(G)|_{p}\leq p^{4}$ if $p\geq 3$ . Moreover, if $G$ is solvable, then $|G:F(G)|_{p}\leq p^{2}$ .