On Chernikov-by-nilpotent groups

TL;DR

This paper investigates groups where the subgroup generated by all conjugates of $ ext{g}^{X_k(G)}$ is Chernikov, proving that the $(k+1)$-th lower central subgroup is also Chernikov with bounded size.

Contribution

It establishes that in such groups, the next lower central subgroup is Chernikov with size bounds depending on initial parameters.

Findings

$ ext{gamma}_{k+1}(G)$ is Chernikov.

Size of $ ext{gamma}_{k+1}(G)$ is $(k,m,n)$-bounded.

Results extend understanding of lower central series in Chernikov-related groups.

Abstract

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 $\langle g^{X_k(G)} \rangle$ 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.