Finite groups in which every two elements generate a soluble subgroup

Abstract

This result was first obtained by John Thompson as a by-product of his classification of N -groups [7]. The proof we present is short and direct. It does not use any classification theorem. The possibility of obtaining results of this type by direct means was first realized by Martin Powell who proved that a finite group in which every three elements generate a soluble subgroup is soluble. An account of his work can be found in [2, pages 473-476]. Powell’s argument uses the Hall-Higman Theorem B. We use a different strategy that we shall now describe. Let G be a soluble group and p an odd prime divisor of |G|. If G contains an abelian p-subgroup that normalizes no nontrivial p′-subgroup of G then Op′(G) = 1 so as G is soluble we must have Op(G) 6= 1. If every abelian p-subgroup of G normalizes a nontrivial p′-subgroup then a result of Thompson [3, Theorem 1.12, page 19] implies that Op′(G) 6= 1. The point of these