Character Tables of Some Selected Groups of Extension Type Using Fischer-Clifford Matrices by R.L. Monaledi A thesis submitted in fulfilment of the requirements for the degree of M.Sc in the Department of Mathematics and Applied Mathematics, University of the Western Cape Supervisor: Prof. R. Fray Co-supervisor: Dr. A.L. Prins December 2015 Declaration I declare that Character Tables of Some Selected Groups of Extension Type Using Fischer- Clifford Matrices is my own work, that it has not been submitted before for any degree or assessment to any other university, and that all the sources that I have used or quoted have been indicated and acknowledged by means of complete references. ........................... R. L. Monaledi December 2015 ii Abstract The aim of this dissertation is to calculate character tables of group extensions. There are several well{developed methods for calculating the character tables of some selected group extensions. The method we study in this dissertation, is a standard application of Clifford theory, made efficient by the use of Fischer-Clifford matrices, as introduced by Fischer. We consider only extensions G of the normal subgroup N by the subgroup G with the property that every irreducible character of N can be extended to an irreducible character of its inertia group in G, if N is abelian. This is indeed the case if G is a split extension, by a well known theorem of Mackey. A brief outline of the classical theory of characters pertinent to this study, is followed by a discussion on the calculation of the conjugacy classes of extension groups by the method of coset analysis. The Clifford theory which provide the basis for the theory of Fischer- Clifford matrices is discussed in detail. Some of the properties of these Fischer-Clifford matrices which make their calculation much easier, are also given. We restrict ourselves to split extension groups G = N:G in which N is always an el- ementary abelian 2-group. In this thesis we are concerned with the construction of the character tables (by means of the technique of Fischer-Clifford matrices) of certain ex- + tension groups which are associated with the orthogonal group O10(2), the automorphism groups U6(2):2, U6(2):3 of the unitary group U6(2) and the smallest Fischer sporadic sim- iii 8 9 9 ple group F i22. These groups are of the type 2 :(U4(2):2), (2 :L3(4)):2, (2 :L3(4)):3 and 6 5 2 :(2 :S6). iv Acknowledgement I would like to bear witness to our Lord Jesus Christ, for being my ultimate frame of reference. I would like to thank Him for all the blessings He bestowed on me. I would like to express my gratitude to the following people for their invaluable support: • my supervisor, Prof R.L. Fray, for his patience and resolute support. It was an honour as well as an immense pleasure being his student. • my co-supervisor Dr A.L. Prins, for the constant encouragement, guidance and advice he gave me during my studies. His readiness to assist and passion for math- ematics has made a huge impact on me. I learnt a lot from working with him. • Pastor Okito, for all the spiritual food in due season and his family for the accom- modation they offered me during my time in Bellville and Kuilsriver. • all the lecturers in the mathematics department at Stellenbosch University, espe- cially those who have taught me at some point in my study career. • my special thanks go to my wife Mary and my children Morepi, Mary-ann and David for all the sacrifices they made and whose love, constant support and encouragement will always be the source of inspiration for me. Special thanks are also due to my family in Hammanskraal especially my mother Dorah who never spared the rod in bringing me up under difficult circumstances. v • very big thank you to my friend Maj L. Ncubukezi for supporting and encouraging me in my studies. • I gratefully acknowledge financial support from the Military Academy at Saldanha especially the transport section for ensuring the availability of transport between the Military Academy and Cape Town. vi Notation and conventions Throughout the thesis all groups will be assumed to be finite, unless otherwise stated. We will use the notation and terminology from the ATLAS [22] and [37]. N natural numbers Z integers Q rational numbers R real numbers C complex numbers G; N; G; H groups 1G the identity element of G. F a field F ∗ F − 0 H ≤ GH is a subgroup of G H E GH is a normal subgroup of G ∼ H = GH is isomorphic to G < x; y > the subgroup generated by x and y N·G an extension of N by G N:G a split extension of N by G o(g) order of g 2 G [g] a conjugacy class of G with representative g NG(H) the normalizer of the subgroup H in G Hg the right coset of H in G X; Y; Ω sets vii jΩj the cardinality of the set Ω Gθ stabilizer of θ in G IG(θ) inertia group of θ in G ng conjugation of n by g g1 s g2 g1 is conjugate to g2 # H the restriction of the character of G to the subgroup H ϕG the induction of the character ϕ of subgroup H to G CG(g) centralizer of g in G Irr(G) set of irreducible characters of G IG the identity character of G χ(G=H) the permutation character of G on H dim(V ) the dimension of a vector space V Sn the symmetric group on n symbols GF (q) the Galois field of q elements V (n; q) a vector space of dimension n over GF (q) Sp(2n; q) symplectic group of dimension 2n over GF (q) Un(q) unitary group viii Contents Declaration ii Abstract iii Acknowledgement v Notation and conventions vii 1 Introduction 1 2 Group Characters 6 2.1 Representations and Characters . 7 2.2 Normal Subgroups . 12 2.3 Restriction of Characters . 14 2.4 Induced Characters . 16 2.5 Permutation Characters . 19 ix 3 Conjugacy Classes of Group Extensions 27 3.1 Extensions of Groups . 27 3.2 Conjugacy classes of semi-direct products. 28 3.3 Conjugacy classes of group extensions (not necessarily split) . 30 4 Clifford Theory 37 4.1 Clifford’s Theorem . 38 5 The Fischer–Clifford matrices 50 5.1 Definitions and Preliminaries . 51 5.2 Properties of Fischer-Clifford matrices . 53 6 On a maximal subgroup of the automorphism group U6(2):2 of U6(2) 59 9 6.1 The group 2 :(L3(4):2) . 60 9 6.2 The conjugacy classes of G = 2 :(L3(4):2) . 64 9 6.3 The inertia groups of G = 2 :(L3(4):2) . 68 6.4 The fusion of H2, H3 and H4 into G ................... 69 9 6.5 The Fischer-Clifford Matrices of 2 :(L3(4):2) . 70 9 6.6 Character Table of 2 :(L3(4):2) . 74 9 6.7 The Fusion of 2 :(L3(4):2) into U6(2):2 . 80 x 7 On a maximal subgroup of the automorphism group U6(2):3 of U6(2) 85 9 7.1 The group 2 :(L3(4):3) . 86 9 7.2 The conjugacy classes of G = 2 :(L3(4):3) . 90 9 7.3 The inertia groups of G = 2 :(L3 (4):3) . 94 7.4 The fusion of H2, H3 and H4 into G ................... 95 9 7.5 The Fischer-Clifford Matrices of 2 :(L3(4):3) . 96 9 7.6 Character Table of 2 :(L3(4):3) . 96 9 7.7 The Fusion of 2 :(L3(4):3) into U6(2):3 . 102 8 + 8 The Fischer-Clifford matrices of 2 :(U4(2):2) as a subgroup of O10(2) 104 8 8.1 On the group 2 :(U4(2):2) and its conjugacy classes . 105 8 8.2 The Inertia groups of 2 :(U4(2):2) . 113 8.3 The Fischer-Clifford Matrices of 28:(U4(2):2) . 114 8 8.4 Character Table of 2 :(U4(2):2) . 116 8 8 8.5 The fusion of 2 :(U4(2):2) into 2 :Sp6(2) . 122 6 9 The Character Table of an inertia group of the maximal subgroup 2 :Sp6(2) of F i22 124 6 5 9.1 The group 2 :(2 :S6) .............................125 6 5 9.2 The conjugacy classes of 2 :(2 :S6) . 129 xi 6 5 9.3 The Inertia groups of 2 :(2 :S6) . 134 5 9.4 The Fusions of the Inertia Factor Groups into 2 :S6 . 135 6 5 9.5 The Fischer-Clifford Matrices of 2 :(2 :S6) . 136 6 5 9.6 Character Table of 2 :(2 :S6) . 138 6 5 6 9.7 The Fusion of 2 :(2 :S6) into 2 :SP6(2) . 165 Bibliography 170 Appendix A 176 Appendix B 180 xii Chapter 1 Introduction The Classification of the Finite Simple Groups (CFSG) emphasizes the importance of the finite simple groups in Group Theory. It is one of the most impressive achievements in the history of Mathematics. The reader is referred to [69] for more literature on the CFSG theorem. Classification of Finite Simple Groups states that each finite simple group is isomorphic to exactly one of the following • A cyclic group of prime order, • An Alternating group An of degree at least 5, • A group of Lie type, • One of twenty{six sporadic simple groups. Since the CFSG, more recent work in group theory has involved methods for the com- putation of character tables of finite groups, and in particular, the character tables of the maximal subgroups of the sporadic simple groups and their automorphism groups.