Schur's theorem and its relation to the closure properties of the non-abelian tensor product

TL;DR

This paper explores the properties of the Schur multiplier and non-abelian tensor products, revealing new insights into their structure and closure properties within specific classes of groups.

Contribution

It establishes that the Schur multiplier of a Noetherian group may not be finitely generated and demonstrates closure properties of non-abelian tensor products for polycyclic groups.

Findings

Schur multiplier of a Noetherian group need not be finitely generated

Non-abelian tensor product of polycyclic groups remains polycyclic

New versions of Schur's theorem are proved

Abstract

We show that the Schur multiplier of a Noetherian group need not be finitely generated. We prove that the non-abelian tensor product of a polycyclic (resp. polycyclic-by-finite) group and a Noetherian group, is a polycyclic (resp. polycyclic-by-finite) group. We also prove new versions of Schur's theorem.