A note on groups with many locally supersoluble subgroups

Abstract

It is proved here that if $G$ is a locally graded group satisfying the minimal condition on subgroups which are not locally supersoluble‎, ‎then $G$ is either locally supersoluble or a Cernikov group‎. ‎The same conclusion holds for locally finite groups satisfying the weak minimal condition on non-(locally supersoluble) subgroups‎. ‎As a consequence‎, ‎it is shown that any infinite locally graded group whose non-(locally supersoluble) subgroups lie into finitely many conjugacy classes must be locally supersoluble.