# Groups, Geometry, and Dynamics

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**Volume 1, Issue 4, 2007, pp. 401–407**

**DOI: 10.4171/GGD/19**

Published online: 2007-12-31

The size of the solvable residual in finite groups

Silvio Dolfi^{[1]}, Marcel Herzog

^{[2]}, Gil Kaplan

^{[3]}and Arieh Lev

^{[4]}(1) Università degli Studi di Firenze, Italy

(2) Tel Aviv University, Israel

(3) The Academic College of Tel Aviv-Yaffo, Israel

(4) The Academic College of Tel-Aviv-Yaffo, Israel

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.

*Keywords: *Commutator subgroup, centre, Frattini subgroup, Fitting subgroup, solvable residual

Dolfi Silvio, Herzog Marcel, Kaplan Gil, Lev Arieh: The size of the solvable residual in finite groups. *Groups Geom. Dyn.* 1 (2007), 401-407. doi: 10.4171/GGD/19