Groups, Geometry, and Dynamics


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Volume 2, Issue 1, 2008, pp. 85–120
DOI: 10.4171/GGD/32

Published online: 2008-03-31

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) Herzen State Pedagogical University, St. Petersburg, Russian Federation
(2) Heinrich-Heine-Universität, Düsseldorf, Germany
(3) Bar-Ilan University, Ramat Gan, Israel
(4) Bar-Ilan University, 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 Nikolai, Grunewald Fritz, Kunyavskii Boris, Plotkin Eugene: A commutator description of the solvable radical of a finite group. Groups Geom. Dyn. 2 (2008), 85-120. doi: 10.4171/GGD/32