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