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 | References | Similar Articles | Additional Information

Abstract: An RA loop is a loop whose loop rings, in characteristic different from , are alternative but not associative. In this paper, we show that every finite subloop of normalized units in the integral loop ring of an RA loop is isomorphic to a subloop of . Moreover, we show that there exist units in the rational loop algebra such that . Thus, a conjecture of Zassenhaus which is open for group rings holds for alternative loop rings (which are not associative).

**1.**R. H. Bruck,*Some results in the theory of linear nonassociative algebras*, Trans. Amer. Math. Soc.**56**(1944), 141--199. MR**6:116b****2.**------,*A survey of binary systems*, Ergeb. Math. Grenzgeb., vol. 20, Springer-Verlag, 1958.MR**20:76****3.**Orin Chein and Edgar G. Goodaire,*Loops whose loop rings are alternative*, Comm. Algebra**14**(1986), no. 2, 293--310.MR**87c:20116****4.**Edgar G. Goodaire,*Alternative loop rings*, Publ. Math. Debrecen**30**(1983), 31--38.MR**85k:20200****5.**Edgar G. Goodaire and César Polcino Milies,*Isomorphisms of integral alternative loop rings*, Rend. Circ. Mat. Palermo**XXXVII**(1988), 126--135.MR**90b:20058****6.**------,*Torsion units in alternative loop rings*, Proc. Amer. Math. Soc.**107**(1989), 7--15.MR**89m:20084****7.**Edgar G. Goodaire and M. M. Parmenter,*Units in alternative loop rings*, Israel J. Math.**53**(1986), no. 2, 209--216.MR**87k:17028****8.**------,*Semi-simplicity of alternative loop rings*, Acta Math. Hungar.**50**(1987), no. 3--4, 241--247.MR**89e:20119****9.**N. Jacobson,*Composition algebras and their automorphisms*, Rend. Circ. Mat. Palermo**VII**(1958), 55--80.MR**21:66****10.**Nathan Jacobson,*Basic algebra I*, W. H. Freeman and Company, San Francisco, 1974.MR**50:9457****11.**H. O. Pflugfelder,*Quasigroups and loops: Introduction*, Heldermann Verlag, Berlin, 1990.

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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