Groups whose proper subgroups of infinite rank have minimax conjugacy classes

If [Formula: see text] is a class of groups, then a group [Formula: see text] is said to be a [Formula: see text]-group, if [Formula: see text] is a [Formula: see text]-group for all [Formula: see text]. This is a generalization of the familiar property of being an [Formula: see text]-group. In the present paper we consider a class [Formula: see text] of soluble-by-finite minimax groups such that [Formula: see text] is a subgroup closed class and if [Formula: see text] is a non-[Formula: see text]-group whose proper subgroups of infinite rank are [Formula: see text]-groups, then there exists a prime [Formula: see text] such that every finite homomorphic image of [Formula: see text] is a cyclic [Formula: see text]-group. Our main result states that if [Formula: see text] is a locally (soluble-by-finite) group of infinite rank which has no simple factor group of infinite rank and if all proper subgroups of [Formula: see text] of infinite rank are [Formula: see text]-groups, then so are all proper subgroups of [Formula: see text]. One can take for [Formula: see text] the class of finite, polycyclic-by-finite, Chernikov, reduced minimax or soluble-by-finite minimax groups.

Finite potent groups

A group is potent if for any element of the group and any prescribed positive integer (dividing its order if this order is finite) there corresponds a finite homomorphic image of the group in which the element has the prescribed integer as its order. The finite potent groups form a finite variety that contains all finite nilpotent groups, all finite metabelian groups, and precisely one simple group, A5.