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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
a_1,a_2,...,a_k the subgroup generated by the elements g, a_iga_i^{-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 b_1,b_2,...,b_l the subgroup generated by the
commutators [g,b_i], 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.