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Proceedings of the American Mathematical Society

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

 

 

Finite subloops of units
in an alternative loop ring


Authors: Edgar G. Goodaire and César Polcino Milies
Journal: Proc. Amer. Math. Soc. 124 (1996), 995-1002
MSC (1991): Primary 17D05; Secondary 16S34, 20N05
DOI: https://doi.org/10.1090/S0002-9939-96-03582-4
MathSciNet review: 1350945
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Abstract: An RA loop is a loop whose loop rings, in characteristic different from $2$, are alternative but not associative. In this paper, we show that every finite subloop $H$ of normalized units in the integral loop ring of an RA loop $L$ is isomorphic to a subloop of $L$. Moreover, we show that there exist units $\gamma_i$ in the rational loop algebra $\mbox{\sf\bf Q}L$ such that $\gamma_k^{-1}(\ldots(\gamma_2^{-1} (\gamma_1^{-1}H\gamma_1)\gamma_2)\ldots) \gamma_k\subseteq L$. Thus, a conjecture of Zassenhaus which is open for group rings holds for alternative loop rings (which are not associative).


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Additional Information

Edgar G. Goodaire
Affiliation: Department of Mathematics, Memorial University of Newfoundland, St. John’s, Newfoundland, Canada A1C 5S7
Email: edgar@math.mun.ca

César Polcino Milies
Affiliation: Universidade de São Paulo, Caixa Postal 20570, 01452-990 São Paulo, Brasil
Email: polcino@ime.usp.br

DOI: https://doi.org/10.1090/S0002-9939-96-03582-4
Received by editor(s): March 23, 1994
Additional Notes: This research was conducted while the first author was a guest of and partially supported by the Instituto de Matemática e Estatística, Universidade de São Paulo, to whom he is most grateful. The research was also supported by FAPESP and CNPq. of Brasil (Proc. No. 93/4440-0 and 501253/91-2, respectively) and the Natural Sciences and Engineering Research Council of Canada, Grant No. 0GP0009087
Communicated by: Lance W. Small
Article copyright: © Copyright 1996 American Mathematical Society