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.
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Nikolai Gordeev, Fritz Grunewald, Boris Kunyavskii, Eugene Plotkin, A commutator description of the solvable radical of a finite group. Groups Geom. Dyn. 2 (2008), no. 1, pp. 85–120DOI 10.4171/GGD/32