Martina Capasso | Searcharxiv

arXiv2026

New variations on the theme of Baer's theorem

Let $\gamma_s(G)$ and $Z_s(G)$ denote the $s$-th terms of the lower and upper central series of a group $G$, respectively. A classical theorem by R. Baer states that if $Z_s(G)$ has finite index $n$ in $G$, then $\gamma_{s+1}(G)$ is also finite. In this paper, we prove that if $G$ is a generalized soluble group such that the quotient $\gamma_s(G)/(\gamma_s(G) \cap Z_t(G))$ has finite rank $r$ for some $s,t$, then the rank of $\gamma_{s+t}(G)$ is finite and $(r,s,t)$-bounded. Moreover, a corresponding result replacing the finite-rank assumption by the condition to be a Chernikov group of bounded size is also obtained. These results extend recent generalizations of the classical Baer's theorem.

math.GR↗

arXiv2025

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

math.GR↗