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THE COVERING NUMBERS of SOME FINITE SIMPLE GROUPS by Michael Epstein a Dissertation Submitted to the Faculty of the Charles E. S

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THE COVERING NUMBERS of SOME FINITE SIMPLE GROUPS by Michael Epstein a Dissertation Submitted to the Faculty of the Charles E. S THE COVERING NUMBERS OF SOME FINITE SIMPLE GROUPS by Michael Epstein A Dissertation Submitted to the Faculty of The Charles E. Schmidt College of Science in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy Florida Atlantic University Boca Raton, FL May 2019 Copyright 2019 by Michael Epstein ii THE COVERING NUMBERS OF SOME FINITE SIMPLE GROUPS by Michael Epstein This dissertation was prepared under the direction of the candidate's dissertation advisor, Dr. Spyros S. Magliveras, Department of Mathematical Sciences, and has been approved by the members of his supervisory committee. It was submitted to the faculty of the Charles E. Schmidt College of Science and was accepted in partial fulfillment of the requirements for the degree of Doctor of Philosophy. SUPERVISORY COMMITTEE: Dissertation Advisor Hari Kalva, Ph.D. D&__~ c2A Lee Kling!~ Rainer Steinwandt, Ph.D. Chair, Department of Mathematical Sci­ ences Fred Richman, Ph.D. ence Khaled Sobhan, Ph.D. Interim Dean, Graduate College iii ACKNOWLEDGEMENTS I am deeply indebted to my advisor Dr. Spyros Magliveras for his guidance, assistance, and endless patience. This dissertation certainly would not exist without his help. I am very fortunate to be able to count myself among his students. I would also like to thank the members of my supervisory committee: Dr. Hari Kalva, Dr. Lee Klingler, and Dr. Fred Richman. In addition I would like to recognize a number of other faculty members from the Department of Mathematical Sciences at FAU that have each contributed to my education in a meaningful way: Dr. Timothy Ford, Dr. William Kalies, Dr. Stephen Locke, Dr. Erik Lundberg, Dr. Yoram Sagher, Dr. Markus Schmidmeier, Dr. Tomas Schonbeck, Dr. Rainer Steinwandt, and Dr. Yuan Wang. I wish to thank my parents, David and Susan Epstein for their encouragement and for always believing in me. Finally, I want to thank my friends and classmates, particularly Yarema Boryshchak, Angela Robinson, Mayra Quiroga, Jessica Khera, Ashley Valentijn, Shaun Miller, Maxime Murray, and most of all Alexandra Milbrand. iv ABSTRACT Author: Michael Epstein Title: The Covering Numbers of some Finite Simple Groups Institution: Florida Atlantic University Dissertation Advisor: Dr. Spyros S. Magliveras Degree: Doctor of Philosophy Year: 2019 A finite cover C of a group G is a finite collection of proper subgroups of G such that G is equal to the union of all of the members of C. Such a cover is called minimal if it has the smallest cardinality among all finite covers of G. The covering number of G, denoted by σ(G), is the number of subgroups in a minimal cover of G. Here we determine the covering numbers of the projective special unitary groups U3(q) for q ≤ 5, and give upper and lower bounds for the covering number of U3(q) when q > 5. We also determine the covering number of the McLaughlin sporadic simple group, and verify previously known results on the covering numbers of the Higman-Sims and Held groups. v To Alex, my love. THE COVERING NUMBERS OF SOME FINITE SIMPLE GROUPS List of Tables .............................. ix 1 Introduction .............................. 1 2 Preliminaries .............................. 4 2.1 Basic Group Theory...........................4 2.2 Covering Numbers of Finite Groups...................6 2.3 Fields................................... 10 2.4 Trace and Semilinear Maps........................ 11 2.5 Conjugate Symmetric Sesquilinear Forms................ 13 2.5.1 Definition and General Properties................ 13 2.5.2 Sesquilinear Forms over Finite Fields.............. 15 2.6 A Bit of Projective Geometry...................... 19 2.6.1 Fundamentals of Projective Geometry.............. 19 2.6.2 Elations and Transvections.................... 22 2.6.3 Unitary Polarities......................... 25 2.7 Unitary Groups.............................. 27 2.7.1 Definition and General Properties................ 27 N 2.7.2 Embedding U3(q) into U3(q ), N odd.............. 29 3 Covering Numbers of Small Projective Special Unitary Groups . 31 3.1 U3(2).................................... 31 3.2 U3(3).................................... 31 3.3 U3(4).................................... 34 vii 3.4 U3(5).................................... 38 4 The Covering Number of U3(q) with q > 5 . 42 4.1 Maximal Subgroups............................ 42 4.2 A Rough Classification of the Elements of G .............. 44 4.2.1 Type 1: Elements Fixing an Absolute Point.......... 44 4.2.2 Sylow p-Subgroups........................ 46 4.2.3 Type 2: Elements Fixing a Nonabsolute Point but no Absolute Points............................... 49 4.2.4 Type 3: Elements Fixing no Points of PG(V )......... 53 4.3 Bounds on the Covering Number.................... 60 5 McLaughlin Group ........................... 66 6 Verification of the covering numbers of the Higman-Sims and Held Groups .................................. 72 6.1 Higman-Sims Group........................... 72 6.2 Held Group................................ 79 7 Conclusion and Open Problems ................... 91 Bibliography .............................. 93 viii LIST OF TABLES 3.1 Conjugacy classes of elements of U3(3)................. 32 3.2 Conjugacy classes of maximal subgroups of U3(3)........... 32 3.3 Incidence matrix for U3(3)........................ 33 3.4 Conjugacy classes of elements of U3(4)................. 35 3.5 Conjugacy classes of maximal subgroups of U3(4)........... 35 3.6 Incidence matrix for U3(4)........................ 35 3.7 Conjugacy classes of elements of U3(5)................. 39 3.8 Conjugacy classes of maximal subgroups of U3(5)........... 39 3.9 Incidence matrix for U3(5)........................ 39 5.1 Conjugacy classes of elements of the McLaughlin group........ 67 5.2 Conjugacy classes of maximal subgroups of the McLaughlin group.. 68 5.3 Incidence matrix for the McLaughlin Group.............. 68 6.1 Conjugacy classes of elements of the Higman-Sims group....... 73 6.2 Conjugacy classes of maximal subgroups of the Higman-Sims group. 74 6.3 Incidence matrix for the Higman-Sims Group.............. 74 6.4 Conjugacy Classes of Elements of the Held Group........... 80 6.5 Conjugacy Classes of Maximal Subgroups of the Held Group..... 81 6.6 Incidence matrix for the Held Group.................. 81 ix CHAPTER 1 INTRODUCTION A cover of a group G is defined to be a collection C of proper subgroups of G such that G = S C. Questions about such covers of groups go back at least as far as the 1950's. It follows from a result of B. H. Neumann (specifically statement (4.4) from [32]) that a group has a finite cover if and only if it has a finite noncyclic homomorphic image. It is natural then to consider the number of subgroups used in a cover of such a group. It is a standard exercise to prove that no group is the union of two proper sub- groups. In 1959 Seymour Haber and Azriel Rosenfeld investigated groups which are the union of three proper subgroups in [18]. They proved that a group is a union of three proper subgroups if and only if it has the Klein 4-group as a homomorphic image. This fact was later reproven in [9] in 1970. This might have been the end of this line of inquiry had J. H. E. Cohn not taken it back up in 1994. In [11] Cohn defines the covering number, denoted σ(G), of a group G which admits a finite cover to be the minimum of the set of cardinalities of the finite covers of G. Cohn then proves that a finite noncyclic supersolvable group G has covering number p + 1 where p is the least prime divisor of G such that G has more than one subgroup of index p. He then determines the covering numbers of A5 and S5, and characterizes the groups with covering number equal to four, five, or six. Cohn also makes two conjectures: first, that a finite noncyclic solvable group G has covering number of the form pa + 1, where p is prime and pa is the order of a chief factor of G, and second, that no group, solvable or otherwise, has covering 1 number equal to seven. Both conjectures were proven in [33] by M. J. Tomkinson, who suggested that covering numbers of simple groups should be studied. Since then there have been a number of results regarding the covering numbers of simple and almost simple groups: In [29], Attila Mar´otiproves several theorems on the covering numbers of alter- nating and symmetric groups. In particular, for the symmetric groups it is shown n−1 that σ(Sn) = 2 if n is odd and n 6= 9 (it was later shown in [25] that this also holds n−2 for n = 9), and σ(Sn) ≤ 2 if n is even. Regarding the alternating groups, Mar´oti n−2 proves that for n 6= 7 or 9, σ(An) ≥ 2 with equality if and only if n ≡ 2 (mod 4) and gives upper bounds in the cases when equality does not hold. Further results on covering numbers of small alternating and symmetric groups can be found in [24, 25, 16]. R. A. Bryce, V. Fedri, and L. Serena investigate the covering numbers of the linear groups GL2(q), SL2(q), P GL2(q), and L2(q) in [10]. Their main result is that if G is any of these groups, then, with a few small exceptions which are handled separately, 1 1 σ(G) = 2 q(q + 1) if q is even and σ(G) = 2 q(q + 1) + 1 if q is odd. J. R. Britnell, A. Evseev, R. M. Guralnick, P. E. Holmes, and A. Mar´oticonsidered the groups GLn(q), SLn(q), P GLn(q), and P SLn(q) for larger n in [2,3], and in particular give the covering numbers of these groups when n ≥ 12. P. E. Holmes determined the covering numbers of several of the sporadic simple groups in [21]. In particular, she proved that σ(M11) = 23, σ(M22) = 771, σ(M23) = 41079, σ(Ly) = 112845655268156, σ(O0N) = 36450855, 24541 ≤ σ(McL) ≤ 24553, and 5165 ≤ σ(J1) ≤ 5415.
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