A commutator description of the solvable radical of a finite group

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 such that for any k elements a1,a2,...,ak 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 such that for any l elements b1,b2,...,bl 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 in each conjugacy class of G every 8 elements generate a solvable subgroup.