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On the nilpotence class of commutative Moufang loops

Published online by Cambridge University Press:  24 October 2008

J. D. H. Smith
J. D. H. Smith
Affiliation:
Technische Hochschule, Darmstadt, West Germany
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Abstract

The nilpotence class of the free commutative Moufang loop on n generators (n > 3) is the maximum allowed by the Bruck-Slaby Theorem, namely n − 1. This is proved by setting up a presentation of an extension of the loop's multiplication group as a nilpotent group of class at most 2n − 2, and then using the Macdonald-Wamsley technique of nilpotent group theory to show that this class is exactly 2n − 2.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 84 , Issue 3 , November 1978 , pp. 387 - 404
Copyright
Copyright © Cambridge Philosophical Society 1978

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References

REFERENCES

(1)Belousov, V. D.O strukture distributivnih kvazigrupp. Mat. Sbornik 50 (1960), 267–298.Google Scholar
(2)Bruck, R. H.A survey of binary systems (Berlin, Springer, 1971).CrossRefGoogle Scholar
(3)Bruck, R. H.An open question concerning Moufang loops. Arch. Math. 10 (1959), 419–421.CrossRefGoogle Scholar
(4)Manin, Ju. I.Kubičeskie formy (Moskva, Nauka, 1972).Google Scholar
(5)Manin, Ju. I. (trans. Hazewinkel, M.). Cubic forms (Amsterdam, North-Holland, 1974).Google Scholar
(6)Smith, J. D. H.Finite distributive quasigroups. Math. Proc. Cambridge Philos. Soc. 80 (1976), 37–41.CrossRefGoogle Scholar
(7)Smith, J. D. H.A second grammar of associators. Math. Proc. Cambridge Philos. Soc. 84 (1978), 405–415.CrossRefGoogle Scholar
(8)Wamsley, J. W. Computation in nilpotent groups (theory). In Proc. 2nd Int. Conf. on the Theory of Groups (Berlin, Springer Lecture Notes, no. 372, 1974), pp. 691–700.Google Scholar