Generalized $FC$-groups with chain conditions

Let $c$ be a positive integer. A group $G$ is called an $FC_c$-group if each element of $G$ has only finitely many conjugates by $\gamma_cG$, and $\gamma_cG$ lies in the $FC$-center of $G$. The $FC_c$-groups with the minimal condition or the maximal conditions on abelian subgroups are investigated and some characterizations of them are obtained. A group is called an $FC_c$-soluble group if it possesses an $FC_c$-series of finite length. Another aim of this article is to give necessary and sufficient conditions for $FC_c$-soluble groups to satisfy the minimal condition or the maximal conditions on abelian subgroups.