Moufang loops of odd order \(pq^3\)
type: Item
instance of: scholarly article
title: Moufang loops of odd order \(pq^3\)
zbMATH Open document ID: 0973.20062
author: Andrew Rajah
published in: Journal of Algebra
publication date: 2001-10-30T00:00:00Z
review text: The author provides a proof for the existence of nonassociative Moufang loops of order \(pq^3\) for every pair of odd primes \(p\) and \(q\) with \(q\equiv 1\pmod p\). He begins his proof by investigating the properties of a nonassociative Moufang loop \(L\) of odd order \(pq^3\), where \(p\) and \(q\) are primes, with \(p<q\). First, it is shown that \(L=CQ\) where \(C\) is a cyclic group of order \(p\), and \(Q\) is a group of order \(q^3\) normal in \(L\). Next, he shows that \(q\equiv 1\pmod p\). Then \(Q\) is proven to be of exponent \(q\), and to be a non-Abelian group if \(p\neq 3\). Finally, a product rule for elements of \(L\) is worked out. Conversely, guided by these properties, a set \(L\) of order \(pq^3\) (\(p\) and \(q\) being odd primes with \(q\equiv 1\pmod p\)) and a product rule for any two elements of \(L\) is defined. A lengthy calculation then shows that \(L\) is indeed a nonassociative Moufang loop.
Mathematics Subject Classification ID: 20N05
zbMATH DE Number: 1563172
zbMATH Keywords: nonassociative Moufang loops
zbMATH Keywords: product rules
MaRDI profile type: Publication
full work available at URL: jabr.2000.8422
OpenAlex ID: W2461632484
cites work: Q5606623
cites work: Contributions to the Theory of Loops
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cites work: On Moufang loops of odd order \(pq^ 2\)
cites work: Moufang loops of odd order \(p_ 1^ 2p_ 2^ 2\cdots p_ m^ 2\)
cites work: Moufang loops of odd order \(p^ 4q_ 1\dots q_ n\)
cites work: Moufang loops of odd order \(p^ \alpha q_ 1^ 2\cdots q_ n^ 2r_ 1\cdots r_ m\)
cites work: Moufang loops of odd order \(p^ mq_ 1\ldots q_ n\)
cites work: The Lattice Point Covering Theorem for Rectangles
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cites work: Corrigendum to: ``On Moufang loops of order the product of three odd primes''
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DOI: 10.1006/JABR.2000.8422
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description: scientific article; zbMATH DE number 1563172