Baer-Suzuki theorem for the solvable radical of a finite group. (Q1009517)

An outline of proof (using the CFSG) is given to the following analogue of a theorem of Baer-Suzuki: Theorem 1.4. Let \(G\) be a finite group. An element \(g\) of prime order \(q>3\) belongs to the solvable radical \(S(G)\) of \(G\) if and only if for any \(x\in G\) the subgroup \(\langle g,x^{-1}gx\rangle\) is solvable. As a corollary, it follows that a finite group \(G\) is solvable if and only if in each conjugacy class of \(G\) every two elements generate a solvable subgroup.