On some closure properties of the non-abelian tensor product and the Bogomolov multiplier

TL;DR

This paper investigates the closure properties of various classes of groups under the non-abelian tensor product, providing conditions for finite generation and analyzing the Bogomolov multiplier of finite simple groups.

Contribution

It establishes closure properties of specific group classes under the non-abelian tensor product and characterizes finite generation conditions.

Findings

Closure of certain group classes under non-abelian tensor product

Necessary and sufficient conditions for finite generation of tensor products

Trivial Bogomolov multiplier for most finite simple groups

Abstract

We prove that the class of nilpotent by finite, solvable by finite, polycyclic by finite, nilpotent of nilpotency class $n$ and supersolvable groups are closed under the formation of the non-abelian tensor product. We provide necessary and sufficient conditions for the non-abelian tensor product of finitely generated groups to be finitely generated. We prove that central extensions of most finite simple groups have trivial Bogomolov multiplier.