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Symmetric Presentations of Non-Abelian Simple Groups California State University, San Bernardino CSUSB ScholarWorks Electronic Theses, Projects, and Dissertations Office of aduateGr Studies 6-2015 SYMMETRIC PRESENTATIONS OF NON-ABELIAN SIMPLE GROUPS Leonard B. Lamp Cal State San Bernardino Follow this and additional works at: https://scholarworks.lib.csusb.edu/etd Part of the Algebra Commons Recommended Citation Lamp, Leonard B., "SYMMETRIC PRESENTATIONS OF NON-ABELIAN SIMPLE GROUPS" (2015). Electronic Theses, Projects, and Dissertations. 222. https://scholarworks.lib.csusb.edu/etd/222 This Thesis is brought to you for free and open access by the Office of aduateGr Studies at CSUSB ScholarWorks. It has been accepted for inclusion in Electronic Theses, Projects, and Dissertations by an authorized administrator of CSUSB ScholarWorks. For more information, please contact [email protected]. Symmetric Presentations of Non-Abelian Simple Groups AThesis Presented to the Faculty of California State University, San Bernardino In Partial Fulfillment of the Requirements for the Degree Master of Arts in Mathematics by Leonard Bo Lamp June 2015 Symmetric Presentations of Non-Abelian Simple Groups AThesis Presented to the Faculty of California State University, San Bernardino by Leonard Bo Lamp June 2015 Approved by: Dr. Zahid Hasan, Committee Chair Date Dr. Gary Griffing, Committee Member Dr. Joseph Chavez, Committee Member Dr. Charles Stanton, Chair, Dr. Corey Dunn Department of Mathematics Graduate Coordinator, Department of Mathematics iii Abstract The goal of this thesis is to show constructions of some of the sporadic groups such as the Mathieu group, M12, the Janko group J1, Projective Special Linear groups, PSL(2, 8), and PSL(2, 11), Unitary group U(3, 3) and many other non-abelian simple groups. Our purpose is to find all simple non-abelian groups as homomorphic images of permutation or monomial progenitors, as well as grasping a deep understanding of group theory. Extension theory is used to determine groups up to isomorphisms. The progenitor, developed by Robert T. Curtis, is an infinite semi-direct product of the following form: P = 2 n : N = ⇡w ⇡ N, w is a reduced word in the t s where 2 n denotes a ⇠ ⇤ { | 2 i0 } ⇤ free product of n copies of the cyclic group of order 2 generated by involutions ti, for 1 i n; and N is a transitive permutation group of degree n which acts on the free product by permuting the involuntary generators by conjugation. Thus we develop methods for factoring by a suitable number of relations in the hope of finding all finite non-abelian simple groups, and in particular one of the 26 sporadic simple groups. Then the algorithm for double coset enumeration together with the First Isomorphism Theorem aids us in proving the homomorphic image of the group we have constructed. After being presented with a group G, we then compute the composition series to solve extension problems. Given a composition such as G = G0 G1 Gn 1 Gn = 1 and the ≥ ≥···≥ − ≥ corresponding factor groups G0/G1 = Q1, ,Gn 2/Gn 1 = Qn 1,Gn 1/Gn = Qn.We ··· − − − − note that G1 = 1, implying Gn 1 = Qn. As we move to the next composition factor − we see that Gn 2/Qn = Qn 1, so that Gn 2 is an extension of Qn 1 by Qn. Following − − − − this procedure we can recapture G from the products of Qi and thus solve the extension problem. The Jordan-Holder theorem then allows us to develop a process to analyze all finite groups. If we knew all finite simple groups and could solve their extension problem, we would arrive at the isomorphism type. We will present how we solve extensions problems while our main focus will lie on extensions that will include the following: semi- direct products, direct products, central extensions and mixed extensions. Lastly, we will discuss Iwasawa’s Lemma and how double coset enumeration aids us in showing the simplicity of some of our groups. iv Acknowledgements I would like to express my deepest appreciation to my immediate adviser, Dr. Hasan for his great interest and inspiration in the pursuit of my studies and in the preparation that led to the development of this thesis. I could not have done it without his commitment and 2 a.m. emails. To Dr. Joseph Chavez and Dr. Gary Griffing I owe a deep sense of gratitude for prompt inspiration, positive attitude, and enthusiasm that enabled me to complete my thesis. I am also extremely thankful for my friends and family for all their understanding and helpful encouragement through my research period, without you I would of been done much earlier. I would like to express a special gratitude and thanks to Dr. Stanton and the entire CSUSB math department for imparting their knowledge and love for mathematics they graciously bestowed upon me. I would also like to express my immeasurable appreciation to my colleague, Dustin Grindsta↵. I could not have finished this thesis without your encouragement and competitive edge. I am deeply grateful for all our lunch breaks, especially when you paid. Best Duo Ever. Lastly, I would like to thank the person that has been there for me throughout my entire education career, Chelsie Lynn Green. Her patience, encouragement, and unwavering love has been the foundation that I have built my life on for the past eight years. Words cannot explain how much of your love and devotion has encouraged me to finish this project, and for that I thank you. v Table of Contents Abstract iii Acknowledgements iv List of Tables viii List of Figures x 1 Introduction 1 1.1 The Origin of the Progenitor . 1 3 1.2 S4 as a Homomorphic Image of 2⇤ : S3 ................... 2 1.2.1 Curtis M24 Example.......................... 2 3 1.2.2 Find a Relation that Produces 2⇤ : S3 ⇠= S4 ............. 3 2 Progenitors and Group Related Preliminaries 4 2.1 Preliminaries .................................. 4 2.2 Examples of Presentations . 8 2.3 Blocks, Transitivity, Primitivity, and Iwasawa’s Lemma . 11 3 Writing Progenitors 14 3.1 WritingRelations................................ 14 3.1.1 Factoring by the Famous Lemma . 14 3.1.2 First Order Relations . 16 3.2 Permutation Progenitors . 17 3.3 Monomial Presentation Progenitors . 18 3.3.1 Character Theory Preliminaries . 18 3.3.2 Orthogonality Relations . 19 3.4 Finding a Monomial Representation . 24 3.5 Wreath Product Progenitors . 29 3.5.1 MethodtoWritingWreathProducts . 35 3.5.2 More Examples of Jesse Train’s Method . 35 3.6 Finding a Permutation Representation for Wreath Products . 36 3.6.1 Using a Simple Loop . 36 3.6.2 Using Classes to Find Permutations of Wreath Products . 37 vi 4 Finite Extensions 40 4.1 Extensions and Related Theorems . 40 4.1.1 Simple Extension Examples . 42 4 4.1.2 Extension Problems Related to the Progenitor 2⇤ : S4 . 51 5 Progenitors with Isomorphism Types 67 3 5.1 7⇤ :m S3 ..................................... 67 9 5.2 2⇤ : D18 ..................................... 68 6 5.3 2⇤ :((C3 C3):C2).............................. 68 7 ⇥ 5.4 2⇤ :(C7 : C3).................................. 69 5 5.5 2⇤ :((C5 : C2):C2) .............................. 69 12 2 5.6 2⇤ :(2 :3) .................................. 70 11 5.7 2⇤ : D22 .................................... 70 10 5.8 2⇤ : D20 .................................... 71 14 5.9 2⇤ : D28 .................................... 71 11 5.10 2⇤ : L2(7) ................................... 72 3 5.11 3⇤ :m S4 ..................................... 72 7 5.12 2⇤ :(7:3) ................................... 73 7 5.13 2⇤ :((7:3):2)................................. 73 2 5.14 7⇤ :m D18 .................................... 73 10 5.15 2⇤ :(2•(5:2):2) ............................... 74 6 5.16 2⇤ :(3•(3:2)) ................................. 74 6 Manual Double Coset Enumeration 75 6.1 Definition for Double Coset Enumeration . 75 5 6.2 Double Coset Enumeration 2⇤ : A5 ..................... 76 6.3 Finding the Center of the Cayley Diagram . 80 7 Double Coset Enumeration over Maximal Subgroups 88 7.1 Double Coset Enumeration of S 2overS . 88 5 ⇥ 4 7.2 Double Coset Enumeration of H over N . 92 7.3 Double Coset Enumeration of G over H . 92 7.4 Computing Double Coset Enumeration of G over N . 93 7.4.1 Mathematical Insight . 94 7.5 Double Coset Enumeration of U(3,3) over a Maximal Subgroup . 94 8 Use of Iwasawa’s Theorem 99 8.1 Use of Iwasawa’s . 99 9 Double Coset Enumeration of L2(8) over D18 104 9.1 Manual Double Coset Enumeration . 104 9.2 Iwasawa’s Lemma to Show G ⇠= L2(8) . 107 9.2.1 G Acts Faithfully X . 107 9.2.2 G Acts Primitively on X . 108 9.2.3 G is Perfect, G0 = G . 108 vii 9.2.4 Conjugates of a Normal Abelian Subgroup K Generate G . 110 9.3 J1 isSimple................................... 111 10 Conclusion 114 Appendix A Double Coset Enumeration Codes 121 A.1 Double Coset Enumeration of PSL(2, 8) over D18 . 121 A.2 Double Coset Enumeration of PSL(2,23) to Find Relations . 125 A.3 Double Coset Enumeration of U(3,3) over a Maximal Subgroup . 130 Appendix B J1 is Simple Using Iwasawa’s 138 Appendix C Solved Composition Factor of the 6• :(PSL(2, 4) : 2) 139 Appendix D Unsuccessful Progenitors and their Relations 143 D.1 WreathProductZ2WrA5 .......................... 143 D.2 WreathProductZ2WrS4........................... 145 D.3 WreathProductZ2WrA4 .......................... 147 D.4 WreathProductZ3WrZ2........................... 149 Appendix E Difficult Extension Problems 150 E.1 Mixed Extension Using Database . 150 E.2 Extension Problem (2 11) :(PGL(2, 11)) . 153 ⇥ • E.3 Extension Problem 2•(U(3, 4) : 2) . 155 Bibliography 157 viii List of Tables 3.1 Classes of A4 .................................. 16 3.2 FirstOrderRelations.............................. 17 3.3 Classes of S3 .................................. 21 3.4 Character Table of Z2 ............................. 21 3.5 Character Table of S3 ............................. 22 3.6 Character Table of S3 ............................. 23 3.7 Character Table of H ............................
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