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Moufang loops with nonnormal commutative centre

Part of: Other generalizations of groups

Published online by Cambridge University Press:  10 January 2020

ALEXANDER N. GRISHKOV  and
ANDREI V. ZAVARNITSINE
ALEXANDER N. GRISHKOV
Affiliation:
Departamento de Matemática, Universidade de São Paulo, Caixa Postal 66281, São Paulo-SP, 05311-970, Brazil. e-mail: [email protected]
ANDREI V. ZAVARNITSINE
Affiliation:
Sobolev Institute of Mathematics, 4, Koptyug av., Novosibirsk, 630090, Russia. e-mail: [email protected]
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Abstract

We construct two infinite series of Moufang loops of exponent 3 whose commutative centre (i. e. the set of elements that commute with all elements of the loop) is not a normal subloop. In particular, we obtain examples of such loops of orders 38 and 311 one of which can be defined as the Moufang triplication of the free Burnside group B(3, 3).

MSC classification

Primary: 20N05: Loops, quasigroups
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 170 , Issue 3 , May 2021 , pp. 609 - 614
Copyright
© Cambridge Philosophical Society 2020

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Footnotes

†

Supported by the program of fundamental scientific research of SB RAS No. I.1.1., project No. 0314-2016-0001 and by FAPESP, process 2017/14489-2.

References

Doro, S.. Simple Moufang loops. Math. Proc. Camb. Phil. Soc. 83, N3 (1978), 377–392.CrossRefGoogle Scholar
Gagola, S. M. III. A Moufang loop’s commutant. Math. Proc. Camb. Phil. Soc. 152, N2 (2012), 193–206.CrossRefGoogle Scholar
Grishkov, A. N. Zavarnitsine, A. V.. Groups with triality. J. Algebra Appl. 5, N4 (2006), 441–463.CrossRefGoogle Scholar
Grishkov, A. N. Zavarnitsine, A. V.. Sylow’s theorem for Moufang loops. J. Algebra. 321, N7 (2009), 1813–1825.CrossRefGoogle Scholar
Grichkov, A. N. Zavarnitsine, A. V.. Multiplication formulas in Moufang loops. Internat. J. Algebra Comput. 26, N4 (2016), 705–725.CrossRefGoogle Scholar
Hall, M. Jr. The Theory of Groups. The Macmillan Co., New York, N.Y, (1959), 434 pp.Google Scholar
Pflugfelder, H. O.. Quasigroups and loops: introduction. Sigma Series in Pure Mathematics, 7 (Berlin: Heldermann Verlag, 1990), 147 pp.Google Scholar