The non-abelian tensor product of normal crossed submodules of groups

Alireza Salemkar,Tahereh Fakhr Taha

doi:10.29252/cgasa.13.1.23

Abstract

In this article, the notions of non-abelian tensor and exterior products of two normal crossed submodules of a given crossed module of groups are introduced and some of their basic properties are established. In particular, we investigate some common properties between normal crossed modules and their tensor products, and present some bounds on the nilpotency class and solvability length of the tensor product, provided such information is given at least on one of the normal crossed submodules.

Highlights

The notion of the non-abelian tensor product of groups was introduced by Brown and Loday [5,6] following ideas of Miller [14], Dennis [8], and has arisen from applications in homotopy theory of a generalized Van Kampen theorem
Fakhr Taha sor products has been to determine which group theoretical properties are closed with respect to forming the tensor product
Pirashvili [19] presented the concept of the tensor product of two abelian crossed modules and investigated its relation to the low-dimensional homology of crossed modules

Summary

Introduction

The notion of the non-abelian tensor product of groups was introduced by Brown and Loday [5,6] following ideas of Miller [14], Dennis [8], and has arisen from applications in homotopy theory of a generalized Van Kampen theorem. Et al [9] showed that the classes of the nilpotent-by-finite, solvable-by-finite, and supersolvable groups are each closed under the formation of the non-abelian tensor product. These results are excellent tools for studying groups. Pirashvili [19] presented the concept of the tensor product of two abelian crossed modules and investigated its relation to the low-dimensional homology of crossed modules. He generalized Whitehead’s universal quadratic functor of abelian groups to abelian crossed modules. We give some of their important properties and study the connection of nilpotency and solvability between the normal crossed modules and their tensor products

Preliminaries on crossed modules

The tensor and exterior products of crossed submodules

On nilpotency and solvability of tensor products