Hostname: page-component-848d4c4894-ttngx Total loading time: 0 Render date: 2024-06-01T22:18:05.027Z Has data issue: false hasContentIssue false
  • English
  • Français

Six Moufang Loops of Units

Published online by Cambridge University Press:  20 November 2018

Edgar G. Goodaire
Edgar G. Goodaire*
Affiliation:
Memorial University of Newfoundland, St. John's, Newfoundland, A1C 5S7, Email: edgar@morgan.ucs.mun.ca
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We compute the loops of units in the integral alternative loop rings of six Moufang loops. Four of these are subloops of the loop of matrices of determinant one in Zorn's vector matrix algebra over a ring of integers while the remaining two are closely related to this interesting algebra. This paper thus serves, in part, to highlight a Moufang analogue of SL(2, Z) which the author suggests is worthy of further study.

Keywords

Primary: 20N0517D05secondary: 16A25
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 44 , Issue 5 , 01 October 1992 , pp. 951 - 973
Copyright
Copyright © Canadian Mathematical Society 1992

References

1. Allen, P.J. and Hobby, C., A characterization of units in Z[A4], J. Algebra 66(1980), 534–543.Google Scholar
2. Allen, P.J. and Hobby, C., A characterization of units in Z[S4], Comm. Algebra (7) 16 (1988), 1479–1505.Google Scholar
3. Chein, Orin, Moufang loops of small order, Mem. Amer. Math. Soc. (13) 197(1978).Google Scholar
4. Chein, Orin and Goodaire, Edgar G., Loops whose loop rings are alternative, Comm. Algebra. 14(1986), 293–310.Google Scholar
5. Goodaire, Edgar G. and Milies, César Polcino, Torsion units in alternative loop rings, Proc. Amer. Math. Soc. 107(1989), 7–15.Google Scholar
6. Goodaire, Edgar G. and Parmenter, M.M., Units in alternative loop rings, Israel J. Math. (2). 53(1986), 209–216.Google Scholar
7. Goodaire, Edgar G. and Parmenter, M.M., Semi-simplicity of alternative loop rings, Acta Math. Hungar. (3-4). 50(1987), 241–247.Google Scholar
8. Graham Higman, The units of group rings, Proc. London Math. Soc. (2) 46(1940), 231–248.Google Scholar
9. Hughes, I. and Pearson, K.R., The group of units of the integral group ring ZS3, Canad. Math. Bull. 15(1972), 529–534.Google Scholar
10. Eric Jespers and Guilherme Leal, A characterization of the unit loop of the integral loop ring ZM i6(Q, 2), to appear in J. Alg.Google Scholar
11. Eric Jespers and Guilherme Leal, Describing units of integral group rings of some 2-groups, Comm. Algebra (6) 19(1991), 1809–1827.Google Scholar
12. Martin Liebeck, W., The classification of finite simple Moufang loops, Math. Proc. Cambridge Philos. Soc.. 102(1987), 33–47.Google Scholar
13. Polcino, C.S. Milies, The group of units of the integral group ring ZD4, Bol. Soc. Brasil. Mat. (2) 4(1973), 85–92.Google Scholar
14. Lowell Paige, J., A class of simple Moufang loops, Proc. Amer. Math. Soc.. 7(1956), 471–482.Google Scholar
15. Parmenter, M.M., Torsion-free normal complements in unit groups of integral group rings, C.R. Math. Rep. Acad. Sci. Canada (4) XII(1990), 113–118.Google Scholar
16. Jiirgen Ritter and Sudarshan Sehgal, K., Construction of units in integral group rings of finite nilpotent groups, Trans. Amer. Math. Soc. (2). 324(1991), 603–621.Google Scholar
17. Sehgal, S.K., Topics in group rings, Marcel Dekker, New York, 1978.Google Scholar
18. Thomas, A.D. and G.V.Wood, Group tables, Shiva Publishing, Orpington, 1980.Google Scholar
19. Zhevlakov, K.A., Slin'ko, A.M., Shestakov, I.P. and Shirshov, A.I., Rings that are nearly associative, Academic Press, New York, 1982, Translated by Harry F. Smith.Google Scholar