symmetric presentations of non-abelian simple groups

168 Pages·2015·1.42 MB·English

CCaalliiffoorrnniiaa SSttaattee UUnniivveerrssiittyy,, SSaann BBeerrnnaarrddiinnoo
CCSSUUSSBB SScchhoollaarrWWoorrkkss
Electronic Theses, Projects, and Dissertations Office of Graduate Studies
6-2015
SSYYMMMMEETTRRIICC PPRREESSEENNTTAATTIIOONNSS OOFF NNOONN--AABBEELLIIAANN SSIIMMPPLLEE
GGRROOUUPPSS
Leonard B. Lamp
Cal State San Bernardino
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Symmetric Presentations of Non-Abelian Simple Groups
A Thesis
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
A Thesis
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 Gri�ng, 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, M , the Janko group J , Projective Special Linear groups,
12 1
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. Extensiontheoryisusedtodeterminegroupsuptoisomorphisms. Theprogenitor,
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 0i } ⇤
free product of n copies of the cyclic group of order 2 generated by involutions t , for
i
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
thealgorithmfordoublecosetenumerationtogetherwiththeFirstIsomorphismTheorem
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 = G G G G = 1 and the
0 1 n 1 n
� �···� � �
corresponding factor groups G /G = Q , ,G /G = Q ,G /G = Q . We
0 1 1 n 2 n 1 n 1 n 1 n n
··· � � � �
note that G = 1, implying G = Q . As we move to the next composition factor
1 n 1 n
�
we see that G /Q = Q , so that G is an extension of Q by Q . Following
n 2 n n 1 n 2 n 1 n
� � � �
this procedure we can recapture G from the products of Q and thus solve the extension
i
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 Gri�ng 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
andhelpfulencouragementthroughmyresearchperiod,withoutyouIwouldofbeendone
muchearlier. IwouldliketoexpressaspecialgratitudeandthankstoDr. Stantonandthe
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
1.2 S as a Homomorphic Image of 2 3 : S . . . . . . . . . . . . . . . . . . . 2
4 ⇤ 3
1.2.1 Curtis M Example . . . . . . . . . . . . . . . . . . . . . . . . . . 2
24
1.2.2 Find a Relation that Produces 2 3 : S = S . . . . . . . . . . . . . 3
⇤ 3 ⇠ 4
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 Writing Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Method to Writing Wreath Products . . . . . . . . . . . . . . . . . 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
4.1.2 Extension Problems Related to the Progenitor 2 4 : S . . . . . . . 51
⇤ 4
5 Progenitors with Isomorphism Types 67
5.1 7 3 : S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
⇤ m 3
5.2 2 9 : D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
⇤ 18
5.3 2 6 : ((C C ) : C ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
⇤ 3 3 2
⇥
5.4 2 7 : (C : C ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
⇤ 7 3
5.5 2 5 : ((C : C ) : C ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
⇤ 5 2 2
5.6 2 12 : (22 : 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
⇤
5.7 2 11 : D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
⇤ 22
5.8 2 10 : D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
⇤ 20
5.9 2 14 : D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
⇤ 28
5.10 2 11 : L (7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
⇤ 2
5.11 3 3 : S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
⇤ m 4
5.12 2 7 : (7 : 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
⇤
5.13 2 7 : ((7 : 3) : 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
⇤
5.14 7 2 : D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
⇤ m 18
5.15 2 10 : (2 (5 : 2) : 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
⇤ •
5.16 2 6 : (3 (3 : 2)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
⇤ •
6 Manual Double Coset Enumeration 75
6.1 Definition for Double Coset Enumeration . . . . . . . . . . . . . . . . . . 75
6.2 Double Coset Enumeration 2 5 : A . . . . . . . . . . . . . . . . . . . . . 76
⇤ 5
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 2 over S . . . . . . . . . . . . . . . . 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 L (8) over D 104
2 18
9.1 Manual Double Coset Enumeration . . . . . . . . . . . . . . . . . . . . . . 104
9.2 Iwasawa’s Lemma to Show G = L (8) . . . . . . . . . . . . . . . . . . . . 107
⇠ 2
9.2.1 G Acts Faithfully X . . . . . . . . . . . . . . . . . . . . . . . . . . 107
9.2.2 G Acts Primitively on X . . . . . . . . . . . . . . . . . . . . . . . 108
9.2.3 G is Perfect, G = G . . . . . . . . . . . . . . . . . . . . . . . . . . 108
0
vii
9.2.4 Conjugates of a Normal Abelian Subgroup K Generate G . . . . . 110
9.3 J is Simple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
1
10 Conclusion 114
Appendix A Double Coset Enumeration Codes 121
A.1 Double Coset Enumeration of PSL(2,8) over D . . . . . . . . . . . . . . 121
18
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 Wreath Product Z2 Wr A5 . . . . . . . . . . . . . . . . . . . . . . . . . . 143
D.2 Wreath Product Z2 Wr S4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
D.3 Wreath Product Z2 Wr A4 . . . . . . . . . . . . . . . . . . . . . . . . . . 147
D.4 Wreath Product Z3 Wr Z2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
Appendix E Di�cult 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 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4
3.2 First Order Relations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.3 Classes of S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3
3.4 Character Table of Z2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.5 Character Table of S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3
3.6 Character Table of S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3
3.7 Character Table of H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.8 Character Table of G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.9 Labeling our ts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
0
3.10 Labeling our ts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
0
3.11 Labeling our ts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
0
3.12 Labeling Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.13 �(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.14 �(4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.15 �(5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.16 �(6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.17 �(7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.18 (3,4)(5,6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
⇤
3.19 (3,7,6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
⇤
5.1 7 3 : S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
⇤ m 3
5.2 2 9 : D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
⇤ 18
5.3 2 6 : ((C C ) : C ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
⇤ 3 3 2
⇥
5.4 2 7 : (C : C ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
⇤ 7 3
5.5 2 5 : ((C : C ) : C ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
⇤ 5 2 2
5.6 2 12 : (22 : 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
⇤
5.7 2 11 : D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
⇤ 22
5.8 2 10 : D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
⇤ 20
5.9 2 14 : D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
⇤ 28
5.10 2 11 : L (7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
⇤ 2
5.11 3 3 : S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
⇤ m 4
5.12 2 7 : (7 : 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
⇤
5.13 2 7 : ((7 : 3) : 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
⇤
ix
5.14 7 2 : D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
⇤ m 18
5.15 2 10 : (2 (5 : 2) : 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
⇤ •
5.16 2 6 : (3 (3 : 2)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
⇤ •
10.1 2 32 : (2 (24 : 5)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
⇤ •
10.2 2 7 : (7 : 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
⇤
10.3 2 7 : (7 : 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
⇤
10.4 2 7 : (7 : 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
⇤
10.5 2 7 : (7 : 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
⇤
10.6 2 5 : (5 : 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
⇤