Some generalisations of Schur's and Baer's theorem and their connection with homological algebra

TL;DR

This paper generalizes Schur's and Baer's theorems in group theory using non-abelian tensor products, extending their applicability to finitely generated groups and exploring related group invariants.

Contribution

It introduces new generalizations of classical theorems in group theory via non-abelian tensor products and applies these to compute group invariants like the $k$-nilpotent multiplier.

Findings

A version of Schur-Baer Theorem for finitely generated groups

Descriptions of the $k$-nilpotent multiplier for $k\\geq 2$

Connections established between classical theorems and homological algebra

Abstract

Schur's Theorem and its generalisation, Baer's Theorem, are distinguished results in group theory, connecting the upper central quotients with the lower central series. The aim of this paper is to generalise these results in two different directions, using novel methods related with the non-abelian tensor product. In particular, we prove a version of Schur-Baer Theorem for finitely generated groups. Then, we apply these newly obtained results to describe the $k$-nilpotent multiplier, for $k\geq 2$, and other invariants of groups.