From Thompson to Baer–Suzuki: A sharp characterization of the solvable radical

We prove that an element g of prime order >3 belongs to the solvable radical R(G) of a finite (or, more generally, a linear) group if and only if for every x∈G the subgroup generated by g, xgx−1 is solvable. This theorem implies that a finite (or a linear) group G is solvable if and only if in each conjugacy class of G every two elements generate a solvable subgroup.