Groups, Geometry, and Dynamics


Full-Text PDF (338 KB) | Table of Contents | GGD summary
Volume 2, Issue 1, 2008, pp. 85–120
DOI: 10.4171/GGD/32

A commutator description of the solvable radical of a finite group

Nikolai Gordeev[1], Fritz Grunewald[2], Boris Kunyavskii[3] and Eugene Plotkin[4]

(1) Department of Mathematics, Herzen State Pedagogical University, 48, Moika Embankment, 191186, ST. PETERSBURG, RUSSIAN FEDERATION
(2) Mathematisches Institut, Heinrich-Heine-Universität, Universitätsstrasse 1, 40225, DÜSSELDORF, GERMANY
(3) Department of Mathematics, Bar-Ilan University, 52900, RAMAT GAN, ISRAEL
(4) Department of Mathematics, Bar-Ilan University, 52900, RAMAT GAN, ISRAEL

We are looking for the smallest integer k > 1 providing the following characterization of the solvable radical R(G) of any finite group G: R(G) coincides with the collection of all gG such that for any k elements a1, a2, … , akG the subgroup generated by the elements g, aigai−1, i = 1, … , k, is solvable. We consider a similar problem of finding the smallest integer ℓ > 1 with the property that R(G) coincides with the collection of all gG such that for any ℓ elements b1, b2, … , bG the subgroup generated by the commutators [g, bi], i = 1, … , ℓ, is solvable. Conjecturally, k = ℓ = 3. We prove that both k and ℓ are at most 7. In particular, this means that a finite group G is solvable if and only if every 8 conjugate elements of G generate a solvable subgroup.

Keywords: Finite group, solvable radical, simple group

Gordeev N, Grunewald F, Kunyavskii B, Plotkin E. A commutator description of the solvable radical of a finite group. Groups Geom. Dyn. 2 (2008), 85-120. doi: 10.4171/GGD/32