Groups with many generalized \(FC\)-subgroup

Abstract

Let \(FC^0\) be the class of all finite groups, and for each non-negative integer \(m\) define by induction the group class \(FC^{m+1}\) consisting of all groups \(G\) such that the factor group \(G/C_G(x^G)\) has the property \(FC^m\) for all elements \(x\) of \(G\). Clearly, \(FC^1\) is the class of \(FC\)-groups and every nilpotent group with class at most \(m\) belongs to \(FC^m\). The class of \(FC^m\)-groups was introduced in [6]. In this article the structure of groups with finitely many normalizers of non-\(FC^m\)-subgroups (respectively, the structure of groups whose subgroups either are subnormal with bounded defect or have the property \(FC^m\)) is investigated.