Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-25T06:01:45.225Z Has data issue: false hasContentIssue false
  • English
  • Français

On Soluble Groups of Automorphism of Riemann Surfaces

Published online by Cambridge University Press:  20 November 2018

Grzegorz Gromadzki
Grzegorz Gromadzki*
Affiliation:
Instytut Matematyki WSP Chodkiewicza 30 85-064 Bydgoszcz Poland and Universidad a Distancia Depto de Matem. Fund. 28040 Madrid Spain
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.

Let G be a soluble group of derived length 3. We show in this paper that if G acts as an automorphism group on a compact Riemann surface of genus g ≠ 3,5,6,10 then it has at most 24(g — 1) elements. Moreover, given a positive integer n we show the existence of a Riemann surface of genus g = n4 + 1 that admits such a group of automorphisms of order 24(g — 1), whilst a surface of specified genus can admit such a group of automorphisms of order 48(g — 1), 40(g — 1), 30(g — 1) and 36(g — 1) respectively.

Keywords

30F9920H1020D45
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 34 , Issue 1 , 01 March 1991 , pp. 67 - 73
Copyright
Copyright © Canadian Mathematical Society 1991

References

1. Accola, R. D. M., On the number of automorphisms of a closed Riemann surface, Trans. Amer. Math. Soc. 131 (1968), 398–404.Google Scholar
2. Chetiya, B. P., Groups of automorphisms of compact Riemann surface. Ph.D. thesis, Birmingham University, 1971.Google Scholar
3. Chetiya, B. P., On genuses of compact Riemann surfaces admitting solvable automorphism groups, Indian J. Pure Appl. Math. 12 (1981), 1312–1318.Google Scholar
4. Chetiya, B. P. and Patra, K., On metabelian groups of automorphisms of compact Riemann surfaces, J. London Math. Soc. 33 (1986), 467–472.Google Scholar
5. Gromadzki, G., Maximal groups of automorphisms of compact Riemann surfaces in various classes of finite groups, Rev. R. Acad. Sc. Madrid,82 (1988), 267–276.Google Scholar
6. Gromadzki, G. and Maclachlan, C., Supersoluble groups of automorphisms of compact Riemann surfaces, Glasgow Math. J.,31 (1989), 321–327.Google Scholar
7. Hurwitz, A., Ùber algebraische gebilde mit eindeutigen Transformationen in sich, Math. Ann. 41 (1893), 403–442.Google Scholar
8. Kulkarni, R., Symmetries of surfaces, Topology 26 (1987), 195–203.Google Scholar
9. Macbeath, A. M., On a theorem of Hurwitz, Proc. Glasgow Math. Assoc. 5 (1961), 90–96.Google Scholar
10. Macbeath, A. M., Residual nilpotency ofFuchsian groups, Illinois J. of Math. 28 (1984), 299–311.Google Scholar
11. Maclachlan, C., Abelian groups of automorphisms of compact Riemann surfaces, Proc. London Math. Soc. 15(3)(1965), 699–712.Google Scholar
12. Maclachlan, C., A bound for the number of automoprhisms of compact Riemann surface, J. London Math. Soc. 44 (1969), 265–272.Google Scholar
13. Singerman, D., Subgroups of Fuchsian groups and finite permutation groups, Bull. London Math. Soc. 2 (1970), 319–332.Google Scholar
14. Zomorrodian, R., Nilpotent automorphism groups of Riemann surfaces, Trans. Amer. Math. Soc. 288 (1985), 241–255.Google Scholar
15. Zomorrodian, R., Clasification of p-groups of automorphisms of Riemann surfaces and their lower central series, Glasgow Math. J. 29 (1987), 237–244.Google Scholar
16. Zomorrodian, R., Bounds for the order of supersoluble automorphism groups of Riemann surfaces, Proc. Amer. Math. Soc. 108 (1990),587-600.Google Scholar