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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



An ``extra'' law for characterizing Moufang loops

Authors: Orin Chein and D. A. Robinson
Journal: Proc. Amer. Math. Soc. 33 (1972), 29-32
MSC: Primary 20N05
MathSciNet review: 0292987
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Abstract: Let $ (G, \cdot )$ be any loop and let $ \lambda, \delta, \alpha $ be mappings of G into G so that $ x\lambda = x \cdot x\delta = x(x\alpha \cdot x)$ for all $ x \in G$. It is shown that the following conditions are equivalent: (a) $ (xy \cdot z)x\alpha = x(y(z \cdot x\alpha ))$ for all $ x,y,z \in G$, (b) $ (G, \cdot )$ is Moufang and $ x\delta $ is in the nucleus of $ (G, \cdot )$ for all $ x \in G$, (c) $ (xy)(z \cdot x\lambda ) = (x \cdot yz)x\lambda $ for all $ x,y,z \in G$. In particular, a loop $ (G, \cdot )$ is extra in that $ (xy \cdot z)x = x(y \cdot zx)$ for all $ x,y,z \in G$ if and only if it satisfies the $ {M_3}$-law in that $ (xy)(z \cdot {x^3}) = (x \cdot yz){x^3}$ for all $ x,y,z \in G$.

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Keywords: Extra loop, Moufang loop, M-loop, nucleus, inverse property, $ {M_\lambda }$-loop
Article copyright: © Copyright 1972 American Mathematical Society