J. G. Thompson [Bull. Am. Math. Soc. 74, 383-437 (1968; Zbl 0159.30804)], using his classification of minimal simple groups, obtains the following criterion of non-solvability: a finite group \(G\) is non-solvable if and only if there are three nontrivial elements \(a,b,c\in G\), whose orders are coprime in pairs, such that \(abc=1\) (this criterion was conjectured by G. A. Miller [Bull. Am. Math. Soc. 19, 303-310 (1913; JFM 44.0167.04)] and independently by P. Hall [J. Lond. Math. Soc. 12, 198-200 (1937; Zbl 0016.39204)]; the authors cite the first item incorrectly).
In the paper under review Thompson’s criterion is sharpened: a finite group \(G\) is non-solvable if and only if there exist three nontrivial elements \(a,b,c\in G\) such that \(a\) is a \(2\)-element, \(b\) is a \(p\)-element where \(p\) is an odd prime number and \(c\) is a \(\{2,p\}'\)-element, satisfying \(abc=1\). As a consequence (Theorem 4) it is proved that a finite group \(G\) is solvable if and only if for any odd prime number \(p\), any \(p\)-element \(g\in G\) and any \(2\)-element \(x\in G\), the group \(\langle g,g^x\rangle\) is solvable.