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Bol Loops of Nilpotence Class Two

Published online by Cambridge University Press:  20 November 2018

Orin Chein  and
Edgar G. Goodaire
Orin Chein
Affiliation:
Temple University, Philadelphia, PA 19122, U.S.A. email: [email protected]
Edgar G. Goodaire
Affiliation:
Memorial University of Newfoundland, St. John's, NF, A1C 5S7 email: [email protected]
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Abstract

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Call a non-Moufang Bol loop minimally non-Moufang if every proper subloop is Moufang and minimally nonassociative if every proper subloop is associative. We prove that these concepts are the same for Bol loops which are nilpotent of class two and in which certain associators square to 1. In the process, we derive many commutator and associator identities which hold in such loops.

Keywords

20N05Bol loopMoufang loopnilpotentcommutatorassociatorminimally nonassociative
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 59 , Issue 2 , 01 April 2007 , pp. 296 - 310
Copyright
Copyright © Canadian Mathematical Society 2007

References

[B] Bol, G., Gewebe und Gruppen. Math. Ann. 114(1937), no. 1, 414–431.Google Scholar
[CG1] Chein, O. and Goodaire, E. G., A new construction of Bol loops of orde. 8k. J. Algebra 287(2005), no. 1, 103–122.Google Scholar
[CG2] Chein, O. and Goodaire, E. G., Moufang loops with a unique nonidentity commutator (associator, square). J. Algebra 130(1990), no. 2, 369–384.Google Scholar
[CG3] Chein, O. and Goodaire, E. G., Minimally nonassociative commutative Moufang loops. Results Math. 39(2001), no. 1-2, 11–17.Google Scholar
[CG4] Chein, O. and Goodaire, E. G., Minimally nonassociative Moufang loops with a unique nonidentity commutator are ring alternative. Comment. Math. Univ. Carolin. 43(2002), no. 1, 1–8.Google Scholar
[CG5] Chein, O. and Goodaire, E. G., Minimally nonassociative nilpotent Moufang loops. J. Algebra 268(2003), no. 1, 327–342.Google Scholar
[CR] Chein, O. and Robinson, D. A., An “extra” law for characterizing Moufang loops. Proc. Amer. Math. Soc. 33(1972), 29–32.Google Scholar
[G] Goodaire, E. G., Units in right alternative loop rings. Publ. Math. Debrecen 59(2001), no. 3–4, 353–362.Google Scholar
[GR1] Goodaire, E. G. and Robinson, D. A., A class of loops with right alternative loop rings. Comm. Algebra 22(1995), no. 14, 5623–5634.Google Scholar
[GR2] Goodaire, E. G. and Robinson, D. A., A construction of loops which admit right alternative loop rings. Results Math. 59(1996), no. 1-2, 56–62.Google Scholar
[KO] Kallaher, M. J. and Ostrom, T. G., Fixed point free linear groups, rank three planes and Bol quasifields. J. Algebra 18(1971), 159–178.Google Scholar
[Ku] Kunen, K., Alternative loop rings. Comm. Algebra 26(1998), no. 2, 557–564.Google Scholar
[MM] Miller, G. A. and Moreno, H. C., Non-abelian groups in which every subgroup is abelian. Trans. Amer. Math. Soc. 4(1903), no. 4, 398–404.Google Scholar
[Pa] Paige, L. J., A theorem on commutative power associative loop algebras. Proc. Amer. Math. Soc. 6(1955), 279–280.Google Scholar
[Pf] Pflugfelder, H. O., Quasigroups and Loops. Introduction. Heldermann Verlag, Berlin, 1990.Google Scholar
[S] Scott, W. R., Group Theory. Prentice-Hall, Inc., 1964.Google Scholar