On Chernikov-by-nilpotent groups

Let ${\gamma }_k=[x_1,\dots ,x_k]$ be the $k$-th lower central group-word. Given a group $G$, we write $X_k(G)$ for the set of ${\gamma }_k$-values and ${\gamma }_k(G)$ for the $k$-th term of the lower central of $G$. This paper deals with groups in which $⟨g^{X_k(G)}⟩$ is a Chernikov group of size at most $(m,n)$ for all $g\in G$. The main result is that ${\gamma }_{k+1}(G)$ is a Chernikov group and its size is $(k,m,n)$-bounded.