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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
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.

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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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