Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-22T12:35:29.033Z Has data issue: false hasContentIssue false
  • English
  • Français

Isomorphisms Between Linear Groups Over Division Rings

Published online by Cambridge University Press:  20 November 2018

Vasilij M. Petechuk
Vasilij M. Petechuk*
Affiliation:
5, Sechenov St. Uzhgorod, Transcarpathia Ukraine
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.

In the present paper we completely describe the isomorphisms between projective elementary groups PSLn and PSLm (n ≥ 2, m ≥ 2) over division rings. It was found that such groups can be isomorphic only if n = m; the division rings are isomorphic or anti-isomorphic, except for the following groups:

PSL(2,F7) and PSL(3,F2); PSL(2, F4) and PSL(2,F5).

The idea is based on a deepening of the classical Hua's approach. This problem has been solved independently by H. Ren, Z. Wan and X. Wu using a different way

Keywords

20G3516A39
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 45 , Issue 5 , 01 October 1993 , pp. 997 - 1008
Copyright
Copyright © Canadian Mathematical Society 1993

References

1. Schreirer, O. and van, B.L. der Waerden, Die Automorphismen der projektiven Gruppen, Abh. Mat. Sem. Hamburg, 6(1928), 303–322.Google Scholar
2. Hahn, A.J., James, D.G. and Weisfeiler, B., Homomorphisms of algebraic and classical groups: a survey, Can. Math. Soc. Conf. Proc. 4(1984), 249–296.Google Scholar
3. Dieudonné, J., La géométrie des groupes classiques, Berlin-Hottingen-Heidelberg, troisième edition, 1971.Google Scholar
4. Hua, L.K. and Wan, C.H., On the automorphisms and isomorphisms of linear groups, J. Chinese Mat. Soc., (1953), 1–32.Google Scholar
5. Hua, L.K. and Wan, C.H., On the automorphisms of the symplectic groups over any field, Ann. of Math. 49(1948), 739–75.Google Scholar
6. Dieudonné, J., On the automorphisms of the classical groups, Mem. Amer. Math. Soc. 2(1951), 1 -95.Google Scholar
7. Ren, H., Automorphisms ofSL2(k) over a class of skew fields, (Chinese), Acta Math. Sin. 424(1981), 566–577.Google Scholar
8. Petechuk, V.M., Isomorphisms of groups that are rich in projective transvections, Mat. Zametki (2) 39 (1986), 186–195.Google Scholar
9. Merzlyakov, Yu. I.,A survey of the latest results on classical group automorphisms. In: The classical group automorphisms, Moscow, “Mir” Publishers, 1976. 250–259.Google Scholar
10. Merzlyakov, Yu. I., The isomorphisms of classical groups over integral rings, Moscow, “Mir” Publishers, 1980.1-272.Google Scholar
11. Petechuk, V.M., The isomorphisms of linear groups over division rings, Lviv, Abstracts of the XIX-th All-Union Algebraic Conference (2) (1987), 220–221.Google Scholar
12. Ren, H., Wan, Z. and Wu, X., Automorphisms and isomorphisms of linear groups over skew fields, Proceedings of Symposia in Pure Mathematics 47(1987), 473–476.Google Scholar
13. Hahn, A.I., Category equivalences and linear groups over rings, J. Algebra 77(1982), 505–543.Google Scholar
14. Hahn, A.I. , Isomorphism theory for orthogonal groups over arbitrary integral domains, J. Algebra (1) 51 (1978), 233–287.Google Scholar