Infinite minimal non-hypercyclic groups

Abstract

If 𝔛 is a class of groups, a group G is minimal non-𝔛 if it is not an 𝔛-group, but all its proper subgroups belong to 𝔛. The aim of this paper is to prove that for an infinite locally graded group, the property of being minimal non-hypercentral and that of being minimal non-hypercyclic are equivalent. Moreover, the main properties of infinite minimal non-hypercentral groups are described. In the last section, we study groups of infinite rank in which all proper subgroups of infinite rank satisfy a generalized supersolubility condition.