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Bounds for the order of supersoluble automorphism groups of Riemann surfaces


Author: Reza Zomorrodian
Journal: Proc. Amer. Math. Soc. 108 (1990), 587-600
MSC: Primary 20H10; Secondary 30F20
DOI: https://doi.org/10.1090/S0002-9939-1990-0994795-X
MathSciNet review: 994795
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Abstract: The maximal automorphism groups of compact Riemann surfaces for a class of groups positioned between nilpotent and soluble groups is investigated. It is proved that if $ G$ is any finite supersoluble group acting as the automorphism group of some compact Riemann surface $ \Omega $ of genus $ g \geq 2$, then:

(i) If $ g = 2$ then $ \vert G\vert \leq 24$ and equality occurs when $ G$ is the supersoluble group $ {D_4} \otimes {{\mathbf{Z}}_3}$ that is the semidirect product of the dihedral group of order 8 and the cyclic group of order 3. This exceptional case occurs when the Fuchsian group $ \Gamma $ has the signature (0;2,4,6), and can cover only this finite supersoluble group of order 24.

(ii) If $ g \geq 3$ then $ \vert G\vert \leq 18\left( {g - 1} \right)$, and if $ \vert G\vert = 18\left( {g - 1} \right)$ then $ \left( {g - 1} \right)$ must be a power of 3. Conversely if $ \left( {g - 1} \right) = {3^n},n \geq 2$, then there is at least one surface $ \Omega $ of genus $ g$ with an automorphism group of order $ 18\left( {g - 1} \right)$ which must be supersoluble since its order is of the form $ 2{3^m}$. This bound corresponds to a specific Fuchsian group given by the signature (0;2,3,18). The terms in the chief series of each of these Fuchsian groups to the point where a torsion-free subgroup is reached are computed.


References [Enhancements On Off] (What's this?)

  • [1] W. Burnside, Theorem of groups of finite order, Note $ K$, Dover, New York, 1955. MR 0069818 (16:1086c)
  • [2] H. S. M. Coxeter, W. O. J. generators and relations for discrete groups, Ergebnisse der Math., Bd. 14, 3rd. ed., Springer, Berlin-Heidelberg-New York, 1972. MR 0349820 (50:2313)
  • [3] K. W. Gruenberg, Residual properties of infinite soluble groups, Proc. London Math. Soc. (3) 7 (1957), 29-62. MR 0087652 (19:386a)
  • [4] R. S. Kulkarni, Symmetries of surfaces, Topology 26 (1987), 195-203. MR 895571 (88m:57051)
  • [5] -, Normal subgroups of Fuchsian groups, Quart. J. of Math. 36 (1985), 325-344. MR 800765 (87c:20089)
  • [6] A. M. Macbeath, On a theorem of Hurwitz, Glasgow Math. J. 5 (1961), 90-96. MR 0146724 (26:4244)
  • [7] -, Residual nilpotency of Fuchsian groups, Illinois J. Math. (2) 28 (1984), 299-311. MR 740620 (85e:30080)
  • [8] C. H. Sah, Groups related to compact Riemann surfaces, Acta Math. 123 (1969), 13-42. MR 0251216 (40:4447)
  • [9] C. L. Siegel, Some remarks on discontinuous groups, Annals of Math. 46 (1945), 708-718. MR 0014088 (7:239c)
  • [10] D. Singerman, Subgroups of Fuchsian groups and finite permutation groups, Bull. London Math. Soc. 2 (1970), 319-323. MR 0281805 (43:7519)
  • [11] M. Weinstein, ed., Between nilpotent and soluble, House, Passaic, New Jersey, 1982, 2-42. MR 655785 (84k:20002)
  • [12] R. Zomorrodian, Nilpotent automorphism groups of Riemann surfaces, Trans. Amer. Math. Soc. 288 (1985), 241-255. MR 773059 (86d:20059)
  • [13] -, Classification of $ p$-groups of automorphisms of Riemann surfaces and their lower central series, Glasgow Math. J. 29 (1987), 237-244. MR 901670 (88j:20050)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1990-0994795-X
Keywords: Fuchsian groups, supersoluble automorphism groups, nilpotent automorphism groups, maximal automorphism groups, compact Riemann surfaces, action of groups, generators, relations, signatures, bounds, chief series
Article copyright: © Copyright 1990 American Mathematical Society

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