Finite subloops of units in an alternative loop ring
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- by Edgar G. Goodaire and César Polcino Milies PDF
- Proc. Amer. Math. Soc. 124 (1996), 995-1002 Request permission
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 $\mathbb {Q}L$ 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).References
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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
- MR Author ID: 140680
- ORCID: 0000-0002-8389-0533
- Email: polcino@ime.usp.br
- 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
- © Copyright 1996 American Mathematical Society
- 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