A characterization of the unit loop of the integral loop ring \({\mathbb{Z}}M_{16}(Q,2)\)
type: Item
instance of: scholarly article
title: A characterization of the unit loop of the integral loop ring \({\mathbb{Z}}M_{16}(Q,2)\)
zbMATH Open document ID: 0772.17011
author: Eric Jespers
author: Guilherme Leal
published in: Journal of Algebra
publication date: 1993-05-16T00:00:00Z
review text: The authors give a concrete description of the loop of units (invertible elements) \(U(\mathbb{Z} L)\) of the integral loop ring \(\mathbb{Z} L\), when \(L=M_{16}(Q,2)\) is the Moufang loop of order 16 that is not the loop of units of the Cayley numbers. In this case it is proved that \(U(\mathbb{Z} L)=\pm LV(\mathbb{Z} L)\), where \(V(\mathbb{Z} L)\) is a loop of integral matrices of determinant one in Zorn's rational vector matrix algebra. \(V(\mathbb{Z} L)\) is also a loop generated by three subgroups each isomorphic to the free group \(V(\mathbb{Z} D_ 4)\) of rank 3, where \(D_ 4\) is the dihedral group of order 8. The results of this work have been generalized and extended in the paper of \textit{E. G. Goodaire} [Can. J. Math. 44, 951-973 (1992)].
Mathematics Subject Classification ID: 17D05
Mathematics Subject Classification ID: 20N05
zbMATH DE Number: 168179
zbMATH Keywords: loop of units
zbMATH Keywords: integral loop ring
zbMATH Keywords: Moufang loop
reviewed by: Elena Broží ková
MaRDI profile type: Publication
full work available at URL: jabr.1993.1032
OpenAlex ID: W2040550960
DOI: 10.1006/JABR.1993.1032
description: scientific article; zbMATH DE number 168179