Groups with many generalized FC-subgroup

Abstract

Let FC⁰ be the class of all finite groups, and for each non-negative integer n define by induction the group class FCⁿ⁺¹ consisting of all groups G such that the factor group G/CG(xG) has the property FCⁿ for all elements x of G. Clearly, FC¹ is the class of FC-groups and every nilpotent group with class at most m belongs to FCm. The class of FCⁿ-groups was introduced in [6]. In this article the structure of groups with finitely many normalizers of non-FCⁿ-subgroups (respectively, the structure of groups whose subgroups either are subnormal with bounded defect or have the property FCⁿ) is investigated.