Large orbits of Hall subgroups of solvable linear groups

Abstract

Suppose that $G$ is a finite solvable group and let $H$ be a Hall $\ddot{I}€$-subgroup, let $b(H)$ be the largest character degree of $H$, we show that $|G:O_{\ddot{I}{€}^{\prime }\ddot{I}€}(G){|}_{\ddot{I}}€\le b(H{)}^2$.