The reduction theorem for relatively maximal subgroups

Abstract

Let [Formula: see text] be a class of finite groups closed under taking subgroups, homomorphic images and extensions. It is known that if [Formula: see text] is a normal subgroup of a finite group [Formula: see text] then the image of an [Formula: see text]-maximal subgroup [Formula: see text] of [Formula: see text] in [Formula: see text] is not, in general, [Formula: see text]-maximal in [Formula: see text]. We say that the reduction [Formula: see text]-theorem holds for a finite group [Formula: see text] if, for every finite group [Formula: see text] that is an extension of [Formula: see text] (i.e. contains [Formula: see text] as a normal subgroup), the number of conjugacy classes of [Formula: see text]-maximal subgroups in [Formula: see text] and [Formula: see text] is the same. The reduction [Formula: see text]-theorem for [Formula: see text] implies that [Formula: see text] is [Formula: see text]-maximal in [Formula: see text] for every extension [Formula: see text] of [Formula: see text] and every [Formula: see text]-maximal subgroup [Formula: see text] of [Formula: see text]. In this paper, we prove that the reduction [Formula: see text]-theorem holds for [Formula: see text] if and only if all [Formula: see text]-maximal subgroups of [Formula: see text] are conjugate in [Formula: see text] and classify the finite groups with this property in terms of composition factors.