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.