A note on theta pairs for maximal subgroups

Abstract

Let M be a maximal subgroup of a finite group G. A pair of subgroups (C, D) of G is called a θ-pair of M if it satisfies the following conditions: (a) D ≤ C and D ⊲ G; (b) (M,C) = G and D ≤ M;(c) C/D has no proper normal subgroup of G/D. A θ-pair (C,D) of M is said to be maximal if M has no θ-pair (C',D') such that C < C'. In this paper we obtain several results on maximal θ-pairs which imply G to be solvable or supersolvable.