Groups with minimal conditions related to finiteness properties on conjugacy classes (Q1825278)

This paper is concerned with conditions related to a group being a minimal non-FC-group. The authors have shown previously that a locally graded group in which every proper subgroup is Chernikov-by-abelian is itself Chernikov-by-abelian [Arch. Math. 51, 193-197 (1988; Zbl 0632.20018)]. Here they show that if \({\mathcal X}\) is the class of groups with Chernikov layers (CL-groups) or the union of the classes of CL- groups and Chernikov-by-abelian groups then a locally graded group in which every proper subgroup is an \({\mathcal X}\)-group is itself an \({\mathcal X}\)-group. From this they deduce results concerning minimal non-FL- groups; that is, groups in which every proper subgroup has finite layers.