A description of Baer–Suzuki type of the solvable radical of a finite group

We obtain the following characterization of the solvable radical R(G)R(G) of any finite group GG: R(G)R(G) coincides with the collection of all g∈Gg∈G such that for any 3 elements a1,a2,a3∈Ga1,a2,a3∈G the subgroup generated by the elements g,aigai−1, i=1,2,3i=1,2,3, is solvable. In particular, this means that a finite group GG is solvable if and only if in each conjugacy class of GG every 4 elements generate a solvable subgroup. The latter result also follows from a theorem of P. Flavell on {2,3}′{2,3}′-elements in the solvable radical of a finite group (which does not use the classification of finite simple groups).