The converse of baer's theorem

TL;DR

This paper proves that if the quotient of a group by its center is finitely generated, then the finiteness of the lower central series quotient implies the finiteness of the next term, providing a converse to Baer's theorem.

Contribution

It establishes a new condition under which the converse of Baer's theorem holds, linking the finite generation of the group modulo its center to the finiteness of certain subgroups.

Findings

If G/Z(G) is finitely generated, then finiteness of G/Z_i(G) implies finiteness of γ_{i+1}(G).

Provides a converse condition to Baer's theorem.

Enhances understanding of the structure of finitely generated groups.

Abstract

The Baer theorem states that for a group $G$ finiteness of $G/Z_i(G)$ implies finiteness of $\gamma_{i+1}(G)$. In this paper we show that if $G/Z(G)$ is finitely generated then the converse is true.