# The size of the solvable residual in finite groups

### Silvio Dolfi

Università degli Studi di Firenze, Italy### Marcel Herzog

Tel Aviv University, Israel### Gil Kaplan

The Academic College of Tel Aviv-Yaffo, Israel### Arieh Lev

The Academic College of Tel-Aviv-Yaffo, Israel

## Abstract

Let G be a finite group. The *solvable residual* of G, denoted by 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 |Res(G)| > |G|β, where

β = log(60)/log(120(24)1/3) ≈ 0.700265861. The constant β in this inequality can not be replaced by a larger constant.

## Cite this article

Silvio Dolfi, Marcel Herzog, Gil Kaplan, Arieh Lev, The size of the solvable residual in finite groups. Groups Geom. Dyn. 1 (2007), no. 4, pp. 401–407

DOI 10.4171/GGD/19