Moufang loops of small order. I
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- by Orin Chein PDF
- Trans. Amer. Math. Soc. 188 (1974), 31-51 Request permission
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.References
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B. Baumslag and B. Chandler, Theory and problems of group theory, Schaum’s Outline Series, McGraw-Hill, New York, 1968.
- Richard Hubert Bruck, A survey of binary systems, Reihe: Gruppentheorie, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1958. MR 0093552, DOI 10.1007/978-3-662-35338-7
- Orin Chein and Hala O. Pflugfelder, On maps $x\rightarrow x^{n}$ and the isotopy-isomorphy property of Moufang loops, Aequationes Math. 6 (1971), 157–161. MR 292986, DOI 10.1007/BF01819747
- Orin Chein and Hala Orlik-Pflugfelder, The smallest Moufang loop, Arch. Math. (Basel) 22 (1971), 573–576. MR 297914, DOI 10.1007/BF01222620
- Orin Chein and D. A. Robinson, An “extra” law for characterizing Moufang loops, Proc. Amer. Math. Soc. 33 (1972), 29–32. MR 292987, DOI 10.1090/S0002-9939-1972-0292987-8
- Ferenc Fenyves, Extra loops. I, Publ. Math. Debrecen 15 (1968), 235–238. MR 237695
- George Glauberman, On loops of odd order. II, J. Algebra 8 (1968), 393–414. MR 222198, DOI 10.1016/0021-8693(68)90050-1
- G. Glauberman and C. R. B. Wright, Nilpotence of finite Moufang $2$-loops, J. Algebra 8 (1968), 415–417. MR 222199, DOI 10.1016/0021-8693(68)90051-3
- Ruth Moufang, Zur Struktur von Alternativkörpern, Math. Ann. 110 (1935), no. 1, 416–430 (German). MR 1512948, DOI 10.1007/BF01448037
- Hala Orlik-Pflugfelder, A special class of Moufang loops, Proc. Amer. Math. Soc. 26 (1970), 583–586. MR 265498, DOI 10.1090/S0002-9939-1970-0265498-1
Additional Information
- © Copyright 1974 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 188 (1974), 31-51
- MSC: Primary 20N05
- DOI: https://doi.org/10.1090/S0002-9947-1974-0330336-3
- MathSciNet review: 0330336