Clifford-Fischer Theory Applied to Certain Groups Associated with Symplectic, Unitary and Thompson

Clifford-Fischer Theory Applied to Certain Groups

Associated with Symplectic, Unitary

and Thompson Groups

By

Ayoub Basheer Mohammed Basheer

[email protected]

[email protected]

Supervisor : Professor Jamshid Moori

[email protected]

School of Mathematics, Statistics and Computer Science

University of KwaZulu-Natal

Pietermaritzburg, South Africa

A thesis submitted in the fulfillment of the requirements for

Philosophiæ Doctor (PhD) in Science at the

School of Mathematics, Statistics and Computer Science, University of

KwaZulu-Natal, Pietermaritzburg

April 2012

i

Abstract

The character table of a finite group is a very powerful tool to study the groups and to prove

many results. Any finite group is either simple or has a normal subgroup and hence will be of

extension type. The classification of finite simple groups, more recent work in group theory, has

been completed in 1985. Researchers turned to look at the maximal subgroups and automorphism

groups of simple groups. The character tables of all the maximal subgroups of the sporadic simple

groups are known, except for some maximal subgroups of the Monster M and the Baby Monster B.

There are several well-developed methods for calculating the character tables of group extensions

and in particular when the kernel of the extension is an elementary abelian group. Character

tables of finite groups can be constructed using various theoretical and computational techniques.

In this thesis we study the method developed by Bernd Fischer and known nowadays as the theory

of Clifford-Fischer matrices. This method derives its fundamentals from the Clifford theory. Let

G = N·G, where N ⊳ G and G/N ∼= G, be a group extension. For each conjugacy class [gi

]G, we

construct a non-singular square matrix Fi

, called a Fischer matrix. Once we have all the Fischer

matrices together with the character tables (ordinary or projective) and fusions of the inertia factor

groups into G, the character table of G is then can be constructed easily. In this thesis we apply

the coset analysis technique (this is a method to find the conjugacy classes of group extensions)

together with theory of Clifford-Fischer matrices to calculate the ordinary character tables of seven

groups of extensions type, in which four are non-split and three are split extensions. These groups

are of the forms: 21+8

+

·A9, 3

7

:Sp(6, 2), 2

6·Sp(6, 2), 2

5·GL(5, 2), 2

10:(U5(2):2), 2

1+6

− :((31+2:8):2)

and 22n·Sp(2n, 2) and 28·Sp(8, 2). In addition we give some general results on the non-split group

2

2n·Sp(2n, 2).

ii

Preface

The work described in this thesis was carried out under the supervision and direction of Professor

Jamshid Moori, School of Mathematics, Statistics and Computer Sciences, University of KwaZulu￾Natal, Pietermaritzburg, from January 2009 to April 2012.

The Thesis represent original work of the author and has not been otherwise been submitted in

any form for any degree or diploma to any University. Where use has been made of the work of

others it is duly acknowledged in the text.

Signature (Student) Date: 5th of April 2012

Signature (Supervisor) Date: 5th of April 2012

iii

Dedication

TO MY PARENTS, MY LOVELY WIFE MUNA, MY LOVELY DAUGHTER FATIMA, MY FAMILY AND TO

THE BEST FRIEND I HAVE EVER GOT MUSA COMTOUR, I DEDICATE THIS WORK.

iv

Acknowledgements

First of all, I thank ALLAH for his Grace and Mercy showered upon me. I heartily express my

profound gratitude to my supervisor, Professor Jamshid Moori, for his invaluable learned guidance,

advises, encouragement, understanding and continued support he has provided me throughout the

duration of my studies which led to the compilation of

• the Postgraduate Diploma project at AIMS - Cape Town 2006,

• the MSc in Mathematics at the University of KwaZulu-Natal 2009,

• this PhD thesis

and hopefully I aim to continue with Prof J. Moori for a Postdoctoral research. I will be always

indebted to him for introducing me to this fascinating area of Mathematics and creating my interest

in Group Theory. Professor Moori is a unique encyclopaedia and I have learnt so much from him,

not only in the academic orientation, but in various walks of life. May ALLAH gives him the power

to enrich furthermore this interesting domain of Mathematics, and in general to advance the wheel

of life forward.

I lovingly thank my precious wife Muna, who supported me each step of the way and without her

help and encouragement it simply never would have been possible to finish this work.

I am grateful for the facilities made available to me by the School of Mathematics, Statistics and

Computer Sciences of the University of KwaZulu-Natal, Pietermaritzburg. I am also grateful for

the financial support that I have received from the University of Khartoum through the Ministry

of Higher Education of Sudan, the National Research Foundation of South Africa (NRF) for a

v

grant holder bursary through Professor Moori and to the University of KwaZulu-Natal for the

graduate assistantship and the doctoral research scholarship for the year 2009. My thanks extend

to the administration of University of Khartoum (UofK), in particular to Mrs Islah Shaaban the

deputy head of teaching assistants and training department, Dr Mohsin the principal of UofK, Dr

Manar the dean to the Faculty of Mathematical Sciences and Dr Eltayib Yousif head of Applied

Mathematics Department at UofK. I would like to thank my officemates Muna Elshareef, T. T.

Seretlo and Kassahun M. Tessema for creating a pleasant working environment.

Finally I sincerely thank my entire extended family represented by Basheer, Suaad, Muna, Fa￾tima, Eihab, Adeeb, Nada, Balla, Hanan, Mujtaba, Ahmed, Khalid, Tayseer, Samah, Rana, Amro,

Ahmed, Mustafa, Iyad, Mohsin and Mohammed.

vi

Table of Contents

Abstract ii

Preface iii

Dedication iv

Acknowledgements v

Table of Contents vii

List of Notations xii

1 Introduction 1

2 Group Extensions 9

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Semidirect Products and Split Extensions . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 The Conjugacy Classes of Group Extensions . . . . . . . . . . . . . . . . . . . . . . . 13

vii

TABLE OF CONTENTS TABLE OF CONTENTS

3 Elementary Theories of Representations and Characters 16

3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 Character Tables and Orthogonality Relations . . . . . . . . . . . . . . . . . . . . . . 19

3.3 Tensor Product of Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.4 Lifting of Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.5 Restriction and Induction of Characters . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.5.1 Restriction of Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.5.2 Induction of Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.6 Permutation Character . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4 Schur Multiplier, Projective Representations and Characters 35

4.1 Schur Multiplier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.2 Projective Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.3 Projective Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5 The Theory of Clifford-Fischer Matrices 46

5.1 The Clifford Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.2 The Fischer Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.3 The Character Tables of Group Extensions . . . . . . . . . . . . . . . . . . . . . . . 54

6 A Group of the Form 3

7

:Sp(6, 2) 57

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

6.2 The Conjugacy Classes of G = 37

:Sp(6, 2) . . . . . . . . . . . . . . . . . . . . . . . . 60

viii

TABLE OF CONTENTS TABLE OF CONTENTS

6.3 Inertia Factor Groups of G = 37

:Sp(6, 2) . . . . . . . . . . . . . . . . . . . . . . . . . 65

6.3.1 First, Second, Third and Fourth Inertia Factor Groups . . . . . . . . . . . . . 67

6.3.2 Fifth and Sixth Inertia Factor Groups . . . . . . . . . . . . . . . . . . . . . . 71

6.3.3 Fusions of Inertia Factor Groups into Sp(6, 2) . . . . . . . . . . . . . . . . . . 75

6.4 Character Tables of the Inertia Factor Groups . . . . . . . . . . . . . . . . . . . . . . 82

6.5 Fischer Matrices of G = 37

:Sp(6, 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.6 The Character Table of G = 37

:Sp(6, 2) . . . . . . . . . . . . . . . . . . . . . . . . . 89

7 Two Maximal Subgroups of Thompson Group Th 95

7.1 Dempwolff Group 25·GL(5, 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

7.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

7.1.2 The Conjugacy Classes of G = 25·GL(5, 2) . . . . . . . . . . . . . . . . . . . . 97

7.1.3 The Inertia Groups of G = 25·GL(5, 2) . . . . . . . . . . . . . . . . . . . . . . 99

7.1.4 Fusion of Classes of H2 into Classes of GL(5, 2) . . . . . . . . . . . . . . . . . 103

7.1.5 Fischer Matrices of 25·GL(5, 2) . . . . . . . . . . . . . . . . . . . . . . . . . . 105

7.1.6 The Character Table of Dempwolff Group G = 25·GL(5, 2) . . . . . . . . . . 108

7.2 A Group of the Form 21+8

+

·A9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

7.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

7.2.2 Conjugacy Classes of Group Extensions and of G = 21+8

+

·A9 . . . . . . . . . . 112

7.2.3 Inertia Factor Groups of G = 21+8

+

·A9 . . . . . . . . . . . . . . . . . . . . . . 115

7.2.4 The Character Table of the Inertia Factor Group H2 . . . . . . . . . . . . . . 115

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TABLE OF CONTENTS TABLE OF CONTENTS

7.2.5 Fusion of the Inertia Factor Groups into A9 . . . . . . . . . . . . . . . . . . . 122

7.2.6 Fischer Matrices of G = 21+8

+

·A9 . . . . . . . . . . . . . . . . . . . . . . . . . 123

7.2.7 The Character Table of G = 21+8

+

·A9 . . . . . . . . . . . . . . . . . . . . . . . 133

8 The Non-Split Extension 2

6·Sp(6, 2) 135

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

8.2 Conjugacy Classes of G = 26·Sp(6, 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

8.3 Inertia Factor Groups of G = 26·Sp(6, 2) . . . . . . . . . . . . . . . . . . . . . . . . . 140

8.4 Fischer Matrices of G = 26·Sp(6, 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

8.5 The Character Table of G = 26·Sp(6, 2) . . . . . . . . . . . . . . . . . . . . . . . . . 147

9 On the Extension 2

2n·Sp(2n, 2) and the Character Table of 2

8·Sp(8, 2) 150

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

9.2 The Group Gn = 22n·Sp(2n, 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

9.3 Generators of the Group G = 28·Sp(8, 2) . . . . . . . . . . . . . . . . . . . . . . . . . 154

9.4 The Conjugacy Classes of G = 28·Sp(8, 2) . . . . . . . . . . . . . . . . . . . . . . . . 157

9.5 The Inertia Factor Groups of G = 28·Sp(8, 2) . . . . . . . . . . . . . . . . . . . . . . 165

9.6 Fischer Matrices of G = 28·Sp(8, 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

9.7 The Character Table of G = 28·Sp(8, 2) . . . . . . . . . . . . . . . . . . . . . . . . . 179

10 Two Groups of the Forms 2

10:(U5(2):2) and 2

1+6

− :((31+2:8):2) 180

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

10.2 Conjugacy Classes of G = 210:(U5(2):2) . . . . . . . . . . . . . . . . . . . . . . . . . 183

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TABLE OF CONTENTS TABLE OF CONTENTS

10.3 Inertia Factor Groups of G = 210:(U5(2):2) . . . . . . . . . . . . . . . . . . . . . . . . 188

10.3.1 The Group H2 = 21+6

− :((31+2:8):2) . . . . . . . . . . . . . . . . . . . . . . . . 191

10.3.2 The Conjugacy Classes of H2 = 21+6

− :((31+2:8):2) . . . . . . . . . . . . . . . . 194

10.3.3 The Inertia Factor Groups of H2 = 21+6

− :((31+2:8):2) . . . . . . . . . . . . . . 196

10.3.4 Fischer Matrices of H2 = 21+6

− :((31+2:8):2) . . . . . . . . . . . . . . . . . . . . 200

10.3.5 The Character Table of H2 = 21+6

− :((31+2:8):2) . . . . . . . . . . . . . . . . . 204

10.4 Fischer Matrices of G = 210:(U5(2):2) . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

10.5 The Character Table of G = 210:(U5(2):2) . . . . . . . . . . . . . . . . . . . . . . . . 213

11 Appendix 215

Bibliography 349

xi

List of Notations

N natural numbers

Z integer numbers

R real numbers

C complex numbers

F field

F

∗ multiplicative group of F

Fq Galois field of q elements

V vector space

V(n, q) vector space of dimension n over the field Fq

dim dimension of a vector space

det determinant of a matrix

tr trace of a matrix

G finite group

e, 1G identity element of G

|G| order of G

o(g) order of g ∈ G

∼= isomorphism of groups

H ≤ G H is a subgroup of G

[G : H] index of H in G

N E G N is a normal subgroup of G

K·Q extension of a group K by a group Q

xii

K × Q direct product of a group K by a group Q

K:Q split extension of K by Q (semidirect product)

K·Q non-split extension of K by Q

G/N quotient group

C1, C2, · · · , Ck distinct conjugacy classes of a finite group G

[g], Cg conjugacy class of g in G

CG(g) centralizer of g ∈ G

Gx stabilizer of x ∈ X when G acts on X

x

G orbit of x ∈ X

|Fix(g)| number of elements in a set X fixed by g ∈ G under the group action

Aut(G) automorphism group of G

G

′

derived or commutator subgroup of G

Z(G) center of G

h

g

conjugation of h by g

M(G) Schur multiplier of G

C(G) representation group or covering group of G

[α] equivalence class of factor sets containing α

H2

(G, M) second cohomology group of a group G with coefficients in M

D2n dihedral group consisting of 2n elements

p

1+2m, p 6= 2 extraspecial p−group of order p

1+2m with center Zp and a quotient

isomorphic to V(2m,p)

2

1+2m

+ extraspecial 2−group of order 21+2m of type “+”

2

1+2m

− extraspecial 2−group of order 21+2m of type “-”

Sylp(G) set of all Sylow p−subgroups of G

Zn group {0, 1, · · · , n − 1} under addition modulo n

Sn symmetric group of n objects

An alternating group of n objects

GL(n, F) general linear group

GL(n, q) finite general linear group

SL(n, F) special linear group

Un(F) unitary group

PGL(n, F) projective general linear group GL(n, F)/Z(GL(n, F))

xiii

PSL(n, F) projective special linear group SL(n, F)/Z(SL(n, F))

Aff(n, F) affine group

Sp(2n, F) symplectic group of dimension 2n over F

Th the sporadic Thompson simple group

Co2 the sporadic Conway group of order 42 305 421 312 000

C(G) algebra of class functions on a finite group G

χ character of a finite group

χρ character afforded by a representation ρ of G

1G trivial character of G

deg degree of a representation or a character

Irr(G) set of the ordinary irreducible characters of G

IrrProj(G, α−1

) set of irreducible projective characters of G with factor set α

(P,α) projective representation of G with factor set α

h , i inner product of class functions or a group generated by two elements

(depends on the context of the discussion)

⊗ tensor product of representations

⊕,

L direct sum

χ↑

G

H character induced from a subgroup H to G

χ↓

G

H character restricted from a group G to its subgroup H

1↑

G

H permutation character of G on the cosets of G/H

inf(χ) inflation (lift) of a character χ

ker φ kernel of a homomorphism φ

Im(φ) image of a function φ

χρ character afforded by a representation ρ of G

Hom(G, A) set of all homomorphisms from G to A

xiv