Infinite minimal non-hypercyclic groups

If {goth X} is a class of groups, a group $G$ is minimal non-{goth X} if it is not an {goth X}-group, but all its proper subgroups belong to {goth 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