Thompson-like characterization of the solvable radical
  We prove that the solvable radical of a finite group G coincides with the set of elements y having the following property: for any x in G the subgroup of G generated by x and y is solvable. We present analogues of this result for finite dimensional Lie algebras and some classes of infinite groups. We also consider a similar problem for pairs of elements. 
R.Guralnick, B.Kunyavskii, E.Plotkin, A.Shalev