Finite groups in which every two elements generate a soluble subgroup
A by-product of \textit{J. G. Thompson}'s \(N\)-groups papers [Bull. Am. Math. Soc. 74, 383-437 (1968; Zbl 0159.30804); Pac. J. Math. 33, 451-536 (1970; Zbl 0256.20024); 39, 483-534 (1971; Zbl 0256.20025); 48, 511-592 (1973; Zbl 0291.20014); 50, 215-297 (1974; Zbl 0291.20015); 51, 573-630 (1974; Zbl 0306.20012)] is the following: Theorem. If \(G\) is a finite group in which each pair of elements generates a solvable subgroup, then \(G\) itself is solvable. The author of the paper under review gives a proof of this result without the use of any classification theorem. This generalizes work of M. Powell [see \textit{N. Blackburn, B. Huppert}, ``Finite Groups II'' (Springer, 1982; Zbl 0477.20001), p. 473-476] who proved the same result under the stronger assumption that every triple of elements in \(G\) generates a solvable subgroup.
scientific article; zbMATH DE number 807304
scientific article