Further Investigations on the Associators of Moufang Loops of Odd Order pqr4
Lois A. Ademola
Department of Mathematics, University of Jos, Jos, Nigeria.
Garba G. Zaku
Department of Mathematics, University of Jos, Jos, Nigeria.
*Author to whom correspondence should be addressed.
Abstract
This paper presents a further investigation on the combination of associators of nonassociative Moufang loop of odd order \(p q r^{4}\), with the conditions that \(p, q\) and \(r\) be primes with \(p<q<r, q \neq 1(\bmod p\) ), \(r \neq 1(\bmod p)\) and \(r \neq 1(\bmod q)\), such that all proper subloops and proper quotient loops of \(L\) are associative. The results \(\left(\mathrm{L}_{\mathrm{a}}, \mathrm{L}_{\mathrm{m}}, \mathrm{L}\right)=\{1\}\) and the extension \(\left(\mathrm{L}_{\mathrm{a}}, \mathrm{L}\langle a\rangle, L\right)=\{1\}\) were obtained on the condition that \(\left|\mathrm{L}_{\mathrm{a}}\right|=\mathrm{r}^{2}\).
Keywords: ABO blood group, Moufang loop, Association of Tuberculosis, order, ABO gene frequencies., prime, nonassociative, normal, subloops