Nilpotent groups of automorphisms of families of Riemann surfaces

1987 ◽

Vol 29 (2) ◽

pp. 237-244 ◽

Cited By ~ 5

Author(s):

Reza Zomorrodian

Keyword(s):

Nilpotent Group ◽

Riemann Surfaces ◽

Lower Central Series ◽

Nilpotent Groups ◽

Fuchsian Groups ◽

Central Series ◽

Local Form ◽

Precise Information ◽

Groups Of Automorphisms ◽

Hurwitz Theorem

In a previous paper [7], I have made a study of the ”nilpotent” analogue of Hurwitz theorem [4] by considering a particular family of signatures called ”nilpotent admissible” [5]. We saw however, that if μN(g) represents the order of the largest nilpotent group of automorphisms of a surface of genus g < 2, then μN(g) < 16(g − 1) and this upper bound occurs when the covering group is a triangle group having the signature (0; 2,4,8) which is in its own 2-local formThe restriction to the nilpotent groups enabled me to obtain much more precise information than was available in the general case. Moreover, all nilpotent groups attaining this maximum order turned out to be ”2-groups”. Since every finite nilpotent group is the direct product of its Sylow subgroups and the groups of automorphisms are factor groups of the Fuchsian groups, it is natural for us to study the Fuchsian groups havin p-local signatures to obtain more precise information about the finite p-groups, and hence about the finite nilpotent groups.This suggests a new problem of determining for each prime p, the “p-group” analogue of Hurwitz theorem. It turns out, as often happens in questions of this nature, that p = 2 and p = 3 are indeed quite exceptional and harder to deal with while computing their lower central series than other primes. Actually, p = 3 is the most difficult, but all the other primes p ≥ 5 can be dealt with at once.

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Supersoluble groups of automorphisms of compact Riemann surfaces

Glasgow Mathematical Journal ◽

10.1017/s0017089500007886 ◽

1989 ◽

Vol 31 (3) ◽

pp. 321-327 ◽

Cited By ~ 4

Author(s):

Grzegorz Gromadzki ◽

Colin MacLachlan

Keyword(s):

Finite Groups ◽

Riemann Surfaces ◽

Nilpotent Groups ◽

Metabelian Groups ◽

Supersoluble Groups ◽

Soluble Groups ◽

Cyclic Groups ◽

Groups Of Automorphisms ◽

Compact Riemann Surfaces ◽

Infinite Sequences

Given an integer g ≥ 2 and a class of finite groups let N(g, ) denote the order of the largest group in that a compact Riemann surface of genus g admits as a group of automorphisms. For the classes of all finite groups, cyclic groups, abelian groups, nilpotent groups, p-groups (given p), soluble groups and finally for metabelian groups, an upper bound for N(g, ) as well as infinite sequences for g for which this bound is attained were found in [5, 6, 7, 8, 13], [4], [10], [15], [16], [1], [2] respectively. This paper deals with that problem for the class of finite supersoluble groups i.e. groups with an invariant series all of whose factors are cyclic. In addition, it goes further by describing exactly those values of g for which the bound is attained. More precisely we prove:

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On nilpotent groups with close groups of automorphisms

Algebra and Logic ◽

10.1007/bf01463202 ◽

1974 ◽

Vol 13 (5) ◽

pp. 306-311 ◽

Cited By ~ 1

Author(s):

G. A. Noskov ◽

V. A. Roman'kov

Keyword(s):

Nilpotent Groups ◽

Groups Of Automorphisms

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Automorphisms of compact non-orientable Riemann surfaces

Glasgow Mathematical Journal ◽

10.1017/s0017089500001142 ◽

1971 ◽

Vol 12 (1) ◽

pp. 50-59 ◽

Cited By ~ 30

Author(s):

D. Singerman

Keyword(s):

Riemann Surface ◽

Automorphism Group ◽

Riemann Surfaces ◽

Group Of Automorphisms ◽

Groups Of Automorphisms ◽

Definition Of ◽

Orientable Surfaces

Using the definition of a Riemann surface, as given for example by Ahlfors and Sario, one can prove that all Riemann surfaces are orientable. However by modifying their definition one can obtain structures on non-orientable surfaces. In fact nonorientable Riemann surfaces have been considered by Klein and Teichmüller amongst others. The problem we consider here is to look for the largest possible groups of automorphisms of compact non-orientable Riemann surfaces and we find that this throws light on the corresponding problem for orientable Riemann surfaces, which was first considered by Hurwitz [1]. He showed that the order of a group of automorphisms of a compact orientable Riemann surface of genus g cannot be bigger than 84(g – 1). This bound he knew to be attained because Klein had exhibited a surface of genus 3 which admitted PSL (2, 7) as its automorphism group, and the order of PSL(2, 7) is 168 = 84(3–1). More recently Macbeath [5, 3] and Lehner and Newman [2] have found infinite families of compact orientable surfaces for which the Hurwitz bound is attained, and in this paper we shall exhibit some new families.

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