Moufang loops of odd order \(pq^3\)
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.
scientific article; zbMATH DE number 1563172