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SPORADIC SIMPLE GROUPS OF LOW GENUS by PESHAWA MOHAMMED KHUDHUR The University of Birmingham School of Mathematics College of Engineering and Physical Science The University of Birmingham December 2015 University of Birmingham Research Archive e-theses repository This unpublished thesis/dissertation is copyright of the author and/or third parties. The intellectual property rights of the author or third parties in respect of this work are as defined by The Copyright Designs and Patents Act 1988 or as modified by any successor legislation. Any use made of information contained in this thesis/dissertation must be in accordance with that legislation and must be properly acknowledged. Further distribution or reproduction in any format is prohibited without the permission of the copyright holder. ABSTRACT Let X be a compact connected Riemann surface of genus g and let f : X ! P1 be a meromorphic function of degree n. Classes of such covers are in one to one correspondence with the primitive systems, which are tuples of elements (x1;x2;··· ;xr) in the symmetric group Sn taken up to conjugation and the action of the braid group, such that x1 ·x2 ·····xr = 1 and G = hx1;x2;··· ;xri is a primitive subgroup G of Sn . This thesis is a contribution to the classification of primitive genus g 6 2 systems of sporadic almost simple groups. ACKNOWLEDGEMENTS I am very thankful to the creator of the universe, the Almighty, and the Merciful for His blessing given during this PhD study. I express my deepest gratitude to my supervisors, Dr. Kay Magaard and Professor Sergey Shpectorov for giving the PhD project. Without their continuing support and guidance, the work presented here may never have been accomplished. I would like to acknowledge the Kurdistan regional government for awarding me a scholarship to pursue the PhD degree in the UK. My special Thanks go to my Kurdish friends at the University of Birmingham, specially, Dr.Rashad Lak, Dr.Sanhan M.Khasraw, Dr.Kawa M.Manmi, and Dr.Haval M. Mohammed. Finally, I am deeply indebted to my family for their everlasting encouragement, and all their financial support especially after my sponsor was no longer able to continue funding my study. CONTENTS 1 Introduction 1 2 Background 7 2.1 Covering and the fundamental group . 7 2.2 Riemann Surfaces and Riemann Existence Theorem . 15 2.3 Hurwitz space . 20 2.4 Primitive Permutation groups . 24 2.5 General criteria for determining possible signatures of ramification types . 26 3 Possible Ramification types for the Sporadic simple groups 35 3.1 Possible Ramification Types . 41 3.1.1 The ClassStructureCharacterTable function . 43 3.1.2 Generating Group Criterion . 44 3.2 The ramification types of Mathieu groups . 44 3.2.1 The Mathieu group M11 ......................... 45 3.2.2 The Mathieu group M12 and its Automorphism group . 46 3.2.3 The Mathieu group M22 and its Automorphism group . 47 3.2.4 The Mathieu group M23 ......................... 49 3.2.5 The Mathieu group M24 ......................... 49 3.3 The ramification types of Janko groups . 50 3.3.1 The Janko group J1 ............................ 50 3.3.2 The Janko group J2 and its Automorphism group . 52 3.3.3 The Janko group J3 and its Automorphism group . 53 3.3.4 The Janko group J4 ............................ 55 3.4 The ramification types of the Conway groups . 55 3.4.1 The Conway group Co3 ......................... 55 3.4.2 The Conway group Co2 and its automorphism group . 56 3.4.3 The Conway group Co1 ......................... 57 3.5 The ramification types of the Higman-Sims group and its automorphism group . 58 3.6 The ramification types of the Fischer Groups . 59 3.6.1 The Fischer group Fi22 and its automorphism group . 59 3.6.2 The Fischer group Fi23 .......................... 60 3.6.3 Fischer group Fi24 and its automorphism group . 62 3.7 The ramification types of the McLaughlin group and its automorphism group . 64 3.8 Ramification types of the Suzuki group and its automorphism group . 65 3.9 The ramification types of the Held group and its automorphism group . 66 3.10 The ramification types of the Rudvalis group and its automorphism group . 68 3.11 The ramification types of the large sporadic simple groups . 69 4 Braid Orbits on Nielsen Classes of Sporadic Simple Groups 74 4.1 MAPCLASS . 76 4.1.1 GeneratingMCOrbits . 76 4.1.2 AllMCOrbits . 79 5 Conclusions 82 CHAPTER 1 INTRODUCTION A connected second countable Hausdorff topological space X together with a complex struc- ture is called a Riemann surface[27]. In other words, a Riemann surface is a one-dimensional complex manifold. Topologically, compact Riemann surfaces are homeomorphic to a sphere, a torus, or a finite number of tori joined together. The genus g of a compact Riemann surface X is defined to be number of tori which are joined together. Let f : X ! P1 be meromorphic function, that is, a holomorphic mapping from X to the Riemann sphere P1. The meromor- − phic function f is of degree n if the fiber f 1(p) for general p 2 P1 is size n. A point a 2 P1 is a branch point if j f −1(a)j < n. Every meromorphic function f has finitely many branch 1 points. Let B = fa1;a2;··· ;arg be the set of branch points of f and let a0 2 P nB. Denote by 1 1 1 P1 = p1(P nB;a0), the fundamental group of P nB with the base point a0. Let gi 2 p1(P nB;a0) be corresponding to the path winding once around point ai in the counter clockwise direction and not around any other branch points. The fundamental group P1 is generated by the ho- motopy classes of the closed paths gi. The homotopy lifting of paths induces an action on the −1 fundamental group P1 on the fiber f (a0) (see Section 2.1). This action gives us a homomor- phism j f from the fundamental group P1 to the symmetric group Sn. The connectedness of the Riemann surface X yields that j f (P1) is a transitive subgroup of Sn and this group is called the monodromy group of the ramified cover f : X ! P1 and is denoted by Mon(X; f ). The gen- erators fg1;:::;grg satisfy the relation g1 · g2 · ::: · gr = 1 and they are distinguished generators of the fundamental group P1. Thus the generators of the monodromy group fg1;:::;grg where 1 gi = j f (gi) will satisfy the same relation . Furthermore, the following statements are satisfied: G = hg1;:::;gri (1) g1 ···gr = 1; (2) r ∑ indgi = 2(n + g − 1) (3) i=1 Equation 3 is called Riemann Hurwitz formula. Here indgi = n−number of orbits of (gi) on −1 G f (a0). Let G =Mon(X; f ) and let Ci = gi be the conjugacy classes of G containing gi. Then the set C = fC1;:::;Crg is the ramification type of f . In our study, we look at the set N(C) = f(g1;:::;gr) : G = hg1;:::;gri; gi ···gr = 1 and gi 2 Ci for all ig which is equivalent to the set of all monodromy homeomorphism and we call the elements of N(C) Nielsen tuples. Let G be a transitive group of Sn. A genus g-system is a tuple (g1;:::;gr) satisfying (1), (2) and (3). If G acts primitively, then the genus g system is called a primitive genus g system. We are more interested in when the meromorphic function f is indecomposable, that is, f can not written of the form f = f1 ◦ f2 where degree of f1 and f2 more than one. Which yields the monodromy group Mon(X; f ) acts primitively on fiber. A natural question is : Which finite groups can be the monodromy groups Mon(X; f ) for a fixed genus of X? In 1990, Guralnick and Thompson [15] put forward the following conjecture: For any fixed non-negative integer g, there is a finite set E (g) of simple groups such that if X is a compact Riemann surfaces of genus g and f : X ! P1 is a meromorphic function, then the non-abelian composition factors of the mononodromy group Mon(X; f ) are either alternating groups or members of E (g). This conjecture was established by Frohardt, Magaard [13]. As E (g) is fi- nite, one would like to determine E (0), E (1) and E (2) explicitly. Moreover, it would be useful for future applications to determine all possibilities of how a group in E (g) can arise, as explic- 2 ∗ S itly as possible. Let E (g) = ( (X; f ) c f (Mon(X; f ))), it is well known that for all X, all prime ∗ ∗ p and all n > 4, Cp 2 E (g) and An 2 E (g) . Indeed for each G which is either a Cp or An, there 1 1 ∼ is a cover y : P ! P depending on G such that Mon(X; f ) = G (see [33, p.17]). We call a group a genus g-group if G = Mon(X; f ) for some (X; f ) and g(X) = g. Using Riemann’s Existence Theorem, Guralnick and Thompson showed in [15] that the ele- ments E (g) occur in a primitive monodromy action, i.e if G 2 E (g), then 9(X; f ) such that G = −1 Mon(X; f ) and G acts primitively on f (a0). This fact brings the Theorem of Aschbacher and Scott into the picture. The Fitting subgroup of the group G is denoted by F(G) and it is defined to be the product of all nilpotent normal subgroups of the group G.
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