JournalsggdVol. 2 , No. 1DOI 10.4171/ggd/32

A commutator description of the solvable radical of a finite group

  • Nikolai Gordeev

    Herzen State Pedagogical University, St. Petersburg, Russian Federation
  • Fritz Grunewald

    Heinrich-Heine-Universität, Düsseldorf, Germany
  • Boris Kunyavskii

    Bar-Ilan University, Ramat Gan, Israel
  • Eugene Plotkin

    Bar-Ilan University, Ramat Gan, Israel
A commutator description of the solvable radical of a finite group cover

Abstract

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 g ∈ G such that for any k elements a1, a2, … , ak ∈ G 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 g ∈ G such that for any ℓ elements b1, b2, … , bℓ ∈ G 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.