Groups without nearly abnormal subgroups

A subgroup M of an infinite group G is said to be nearly maximal if it is a maximal element of the set of all subgroups of G having infinite index; i.e. if the index |G:M| is infinite but every subgroup of G properly containing M has finite index in G. The near Frattini subgroup ψ(G) of an infinite group G can now be defined as the intersection of all nearly maximal subgroups of G, with the stipulation that ψ(G)=G if G has no nearly maximal subgroups. These concepts have been introduced by Riles [5]. It was later proved by Lennox and Robinson [4] that a finitely generated soluble-by-finite group G is infinite-by-nilpotent if and only if all its nearly maximal subgroups are normal. It follows that in the class of finitely generated soluble-by-finite groups the property of being finite-by-nilpotent is inherited from the near Frattini factor group G/ψ(G) to the group G itself. In the study of ordinary Frattini properties of infinite groups, some analogies exist between the behaviour of finitely generated soluble groups and soluble minimax residually finite groups (see for instance [6] and [7]). This fact could suggest that a result corresponding to that of Lennox and Robinson also holds for soluble residually finite minimax groups. Unfortunately in this case the property of being finite-by-nilpotent cannot be detected from the behaviour of nearly maximal subgroups, this phenomenon depending on the fact that infinite soluble residually finite minimax groups may be poor of such subgroups.