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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Moufang loops of small order. I


Author: Orin Chein
Journal: Trans. Amer. Math. Soc. 188 (1974), 31-51
MSC: Primary 20N05
MathSciNet review: 0330336
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Abstract: The main result of this paper is the determination of all nonassociative Moufang loops of orders $ \leq 31$. Combinatorial type methods are used to consider a number of cases which lead to the discovery of 13 loops of the type in question and prove that there can be no others. All of the loops found are isomorphic to all of their loop isotopes, are solvable, and satisfy both Lagrange's theorem and Sylow's main theorem.

In addition to finding the loops referred to above, we prove that Moufang loops of orders p, $ {p^2}$, $ {p^3}$ or pq (for p and q prime) must be groups.

Finally, a method is found for constructing nonassociative Moufang loops as extensions of nonabelian groups by the cyclic group of order 2.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1974-0330336-3
PII: S 0002-9947(1974)0330336-3
Keywords: Moufang loop, diassociative, subloop, order, Lagrange's theorem, Sylow's theorem, center, nucleus
Article copyright: © Copyright 1974 American Mathematical Society