The size of the solvable residual in finite groups

Abstract

Let $G$ be a finite group. The solvable residual of $G$, denoted by $\mathrm{Res}(G)$, is the smallest normal subgroup of $G$ such that the respective quotient is solvable. We prove that every finite non-trivial group $G$ with a trivial Fitting subgroup satisfies the inequality $|\mathrm{Res}(G)|>|G{|}^{\beta }$, where $\beta =\log (60)/\log (120(24{)}^{1/3})\approx 0.700265861$. The constant $\beta$ in this inequality can not be replaced by a larger constant.