Certain Group Extensions Associated with M
22:2, M
24and HS:2
D. S Chikopela
25745883
Dissertation submitted in fulfilment of the requirements for the degree Master of Science in Mathematics at the Mafikeng Campus of the North-West
University
Supervisor: Prof. T. T Seretlo
Co-supervisor: Dr. T. T Le
Character tables of finite groups provide substantial information about a group and their use is of great importance in physical sciences and pure mathematics. Any finite group is either simple or has a proper normal subgroup and hence may be of extension type ¯G = N·G. There are several methods
for constructing character tables of group extensions especially when the kernel of the extension, N is an elementary abelian p-group. In this dissertation, we use a more natural approach to the study of the character table of ¯G called the Clifford-Fischer matrices due to Bernd Fischer. This method is based on Clifford’s Theory. For each conjugacy class [g]G, we construct an invertible
matrix M (g), called a Fischer matrix. We have employed in this dissertation a new approach of calculating these Fischer matrices. For the determination of the conjugacy classes, we use coset analysis developed by Moori. Having all the conjugacy classes, Fischer matrices, character tables and fusions of the inertia factor groups into G, we can easily construct the character table of ¯G. We will apply the method of Fischer matrices to construct character tables of group extensions associated with the full automorphism group of the Mathieu group, M22denoted by M22:2, the full
automorphism group of the Higman-Sims group, HS denoted by HS:2 and the largest of the five Mathieu groups, M24. These groups are 24:S6 a maximal subgroup of M22:2, a subgroup 25:A6 of
The work described in this dissertation was carried out under the supervision of Prof. T. T Seretlo and co-supervision of Dr. T. T Le, Department of Mathematical Sciences, North-West University, Mafikeng, from July 2015 to September 2016.
The dissertation represents original work by the author and has not otherwise been submitted in any form for any degree or diploma to any other university. Where use has been made of the work of others it is duly acknowledged in the text.
Signed:
... D. S Chikopela (Student)
... Prof. T. T Seretlo (Supervisor)
... Dr. T. T Le (Co-supervisor)
To my two daughters Taizya and Kuzanga, my wife Charity and my parents, this is as a result of your sacrifice and I dedicate this work to you.
I am deeply and overall very indebted to my supervisor, Prof. T. T Seretlo, for the guidance, support and encouragements he gave me during my studies. It is not easy to express in words my deep felt gratitude having worked under his supervision. To my co-supervisor Dr. T. T Le, no words can justifiably express my gratitude for tutoring, monitoring and giving me extremely helpful advice in mathematics.
Financial support from NWU (Mafikeng) is acknowledged.
To my friends Chrisper Chileshe, Hope Sabao and Mainza Mbokoma thank you for the memorable time we shared together in the office and for being such helpful and good friends.
I would also like to thank the academic staff of the Department of Mathematical Sciences at the North-West University (Mafikeng) for the conducive environment offered during the research period as well as the opportunity granted to me to serve as a student assistant in the department.
Our notation is standard and we follow Alperin [3], Atlas [10] and Atlas V3 [42].
N natural numbers Z integers
C complex numbers V vector space
dim dimension of a vector space F field
F∗ multiplicative group of F Fq finite field of q elements
G, N, M, K finite groups N ≤ G N is a subgroup of G [G : N ] the index of N in G N E G N is a normal subgroup of G N.G group extension N :G split extension N·G non-split extension
M × N direct product of the groups M and N Zm cyclic group of order m
Sn symmetric group on the set {1, 2, .., n}
An alternating group on the set {1, 2, .., n}
N g right coset of N containing g G/N quotient group
NG(K) normalizer of K in G
o(g) order of g ∈ G CG(g) centralizer of g ∈ G
¯
H inertia group of ¯G H inertia factor group of ¯G tr trace of a matrix
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 [g]G the conjugacy class of g ∈ G
g1 ∼ g2 g1 is conjugate to g2 in G
Aut(G) automorphism group of G G0 commutator subgroup of G Z(G) center of G
M (G) Schur multiplier of G
GL(n, F) general linear group of degree n over F GL(n, q) finite general linear group of degree n over Fq
hχ, φi inner product of class functions ⊗ tensor product of representations ⊕ direct sum
χ↑GN character induced from a subgroup N to G
χ↓GN character restricted from the main group G to a subgroup N 1↑GN permutation character of G on the cosets of G/N
ker(ϕ) kernel of a homomorphism ϕ Im(ϕ) image of a function ϕ
HS the Higman-Sims group
HS:2 full automorphism of the Higman-Sims group 2n an elementary abelian group of order 2n
2.1 General Form of a Character Table . . . 15
5.1 Mathieu Groups and Steiner Systems . . . 50
5.2 Generators of M22:2 . . . 52
5.3 Generators of 24:S6 . . . 53
5.4 Generators of 24 . . . 53
5.5 Generators of S6 . . . 54
5.6 Fixed Points for the Action of S6 on 24 . . . 54
5.7 Conjugacy Classes of 24:S6 . . . 56
5.8 Character Table of H1 = S6 . . . 58
5.9 Character Table of H2 = 2 × S4 . . . 58
5.10 Fusion of the Classes from 2 × S4 to S6 . . . 58
5.11 Character Table of 24:S6 . . . 63
5.12 The Character Table of M22:2 . . . 65
5.13 Computation for the Fusion of 24:S6 into M22:2 . . . 69
5.14 Summary for the Fusion of 24:S6 into M22:2 . . . 70
6.1 Fixed Points for the Action of A6 on 25 . . . 74
6.2 Conjugacy Classes of 25:A6 . . . 75
6.3 Generators of G = A6 . . . 76
6.4 Character Table of H1 = H2 = A6 . . . 76
6.5 Character Table of H3 = H4 = S4 . . . 77
6.6 Fusion of the Classes from S4 to A6 . . . 77
6.7 Character Table of 25:A6 . . . 82
6.8 Computation for the Fusion of 25:A6 into 25:S6 . . . 87
6.9 Summary for the Fusion of 25:A6 into 25:S6 . . . 88
7.1 Generators of 26:(3·S6) . . . 90
7.2 11 × 11 Matrix Generators of 3·S6 . . . 90
7.3 6 × 6 Matrix Generators of 3·S6 . . . 91
7.6 Character Table of H1 = 3·S6 . . . 93
7.7 Character Table of H2 = S5 . . . 94
7.8 Character Table of H3 = 2 × S4 . . . 94
7.9 Fusion of the Classes from S5 to 3·S6 . . . 94
7.10 Fusion of the Classes from 2 × S4 to 3·S6 . . . 94
7.11 Fischer Matrices of 26:(3·S6) . . . 95
7.12 Character Table of 26:(3·S6) . . . 97
7.13 Computation for the Fusion of 26:(3·S6) into M24 . . . 103
7.14 Summary for the Fusion of 26:(3·S6) into M24 . . . 104
A.1 Character Table of 25:S 6 . . . 109
Abstract i
Preface ii
Dedication iii
Acknowledgements iv
List of Notations v
List of Tables vii
1 Introduction 1
2 Representation and Character Theories 5
2.1 Some Review on Group Theory . . . 5
2.2 Representations and Characters . . . 8
2.2.1 Inner product and orthogonality relations . . . 11
2.2.2 Character tables . . . 14
2.3 Normal Subgroups and Lifted Characters . . . 15
2.3.1 Lifted characters . . . 16
2.3.2 Normal subgroups . . . 17
2.4 Products of Characters . . . 18
2.5 Restriction and Induction of Characters . . . 20
2.5.1 Restriction to a subgroup . . . 20
2.5.2 Induction to the main group . . . 21
2.6 Permutation Character . . . 23
3 Group Extensions 27 3.1 Definitions and Basic Results . . . 27
3.2 Semi-Direct Products and Split Extensions . . . 29
4.1 Clifford’s Theory . . . 33
4.2 Clifford-Fischer Matrices . . . 37
4.2.1 Definitions and notations . . . 37
4.2.2 Properties of Fischer matrices . . . 39
4.2.3 Further properties of Fischer matrices . . . 41
4.3 Projective Representations and Characters . . . 44
4.3.1 Construction of projective representations and characters . . . 46
4.3.2 Relating projective characters to Clifford-Fischer Theory . . . 47
5 On a Maximal Subgroup of M22:2 49 5.1 The Mathieu Groups . . . 49
5.1.1 Construction of the Mathieu groups . . . 50
5.1.2 The group M22:2 . . . 51 5.2 The Group ¯G = 24:S6 . . . 52 5.2.1 Construction of N = 24 and G = S 6 . . . 53 5.2.2 The action of S6 on 24 . . . 54 5.3 Conjugacy Classes of 24:S6 . . . 54
5.4 The Fischer Matrices of 24:S6 . . . 57
5.4.1 The character table of ¯G = 24:S6 . . . 61
5.5 Fusion of 24:S 6 into M22:2 . . . 64
6 The Group 25:A6 in HS:2 71 6.1 The Higman-Sims Group . . . 71
6.1.1 The group HS:2 . . . 72
6.2 The Group 25:A 6 . . . 72
6.2.1 Construction of N = 25 and G = A6 . . . 73
6.2.2 The action of A6 on 25 . . . 73
6.3 The Conjugacy Classes of ¯G = 25:A6 . . . 74
6.4 The Action of ¯G = 25:A6 on Irr(25) . . . 75
6.5 The Fischer Matrices of ¯G = 25:A6 . . . 77
6.5.1 Character table of ¯G = 25:A6 . . . 81 6.6 Fusion of 25:A6 into 25:S6 . . . 84 7 On a Maximal Subgroup of M24 89 7.1 The Group 26:(3·S 6) . . . 89 7.1.1 Construction of N = 26 and G = 3·S6 . . . 90
7.2 The Action of G = 3:S6 on 26 and Irr(26) . . . 90
7.2.1 The conjugacy classes of ¯G = 26:(3·S6) . . . 92
7.4 Fusion of ¯G = 26:(3·S6) into M24 . . . 99
A Appendix 105
A.1 GAP Programmes . . . 105 A.2 Character Tables . . . 108
1
Introduction
Every finite simple group is isomorphic to one of the following: the 26 sporadic simple groups, cyclic groups of prime order, alternating groups An (n > 4, n ∈ Z) and finite simple groups of
Lie type. We consider in this dissertation, groups associated with some sporadic simple groups and their full automorphism groups. We aim at constructing character tables of these groups as they contain important information about them. All the groups discussed in this dissertation are of extension type, ¯G = N·G, where N is a normal subgroup of ¯G and G is such that ¯G/N ∼= G. To
determine the character tables of group extensions, we use the method of Clifford-Fischer matrices. This method is due to Bernd Fischer [11] and is mainly based on Clifford’s Theory. We denote the split and non-split extensions by ¯G = N :G and ¯G = N·G respectively. The method then involves determining the orbits of the action of ¯G on the irreducible characters of N , denoted by Irr(N ). This action is given by θx(g) = θ(xgx−1), for all x, g ∈ ¯G and θ ∈ Irr(N ) . For each θ ∈ Irr(N ), the stabilizer of θ is then determined. The whole method is as a result of the one-to-one correspondence between the set of specific irreducible characters (with some conditions) of such stabilizers which are the inertia groups and the set of irreducible characters (in like manner with certain conditions) of ¯G (see Theorem 4.1.5). We induce such characters of the inertia groups to the main group ¯G. This will lead to the construction of the character table of ¯G by using the character tables of G and those of the subgroups of G called inertia factor groups.
The first step in constructing the character table of any finite group is to find its conjugacy classes. If ¯G = N·G is an extension, the technique of coset analysis developed by Moori [26] can be used to
calculate the conjugacy classes of ¯G. The idea of this technique is to consider for each conjugacy class [g]G, one coset N ¯g, where ¯g is a pre-image of g ∈ G. Corresponding to this class, we construct a
number of conjugacy classes of ¯G. In this dissertation, we have applied the coset analysis technique together with the theory of Clifford-Fischer matrices to three groups of extension type associated with the full automorphism group of M22 denoted by M22:2, the full automorphism group of the
Higman-Sims group, HS denoted by HS:2 and the largest of the five Mathieu groups, M24. This
dissertation is composed of 7 chapters and an Appendix. In Appendix, we present some GAP [14] programmes and character tables. We next give some details on the structures of Chapters 2 to 7.
In Chapter 2, we discuss some preliminary results on representations and characters of finite groups. Section 2.1 is a review of some important results in group theory. Some definitions and basic results on representations and characters are given in Section 2.2. The characters of direct products of groups and factor groups are discussed in Sections 2.3 and 2.4. In Section 2.5, we establish some relationship between the characters of a group and its subgroup by introducing the notions of restriction and induction of characters. To achieve this, a proof of the Frobenius Reciprocity Theorem is given from Isaacs [18]. We conclude Chapter 2 with Section 2.6 on the theory of permutation characters. We shall use permutation characters to determine the (partial) fusions of conjugacy classes of subgroups into the main group in Chapters 5, 6 and 7.
Chapter 3 discusses the general theory of group extensions and is divided into three sections. In Section 3.1, we introduce notations, give definitions and present basic results on group extensions. In Section 3.2, we discuss a result from Scott [39] that every split extension ¯G of N by G, where N is a normal subgroup of ¯G, is identical to a semi-direct product of N by G. We also show that every semi-direct product ¯G of N by G realizes a homomorphism θ : G −→ Aut(N ). In Section 3.3, we discuss the technique of coset analysis for computing conjugacy classes of group extensions. Coset analysis is applicable to both split and non-split extensions. This technique was developed by Moori in [26], [27] and it has been widely used in the computation of the conjugacy classes of some maximal subgroups of finite simple groups and their automorphism groups (see for example Ali [1], Seretlo [33], Mpono [34] and Seretlo [40]).
In Chapter 4, we study the construction of character tables of group extensions ¯G of N by G, where N is a normal subgroup of ¯G but is not necessarily abelian. We first discuss Clifford’s Theory in Section 4.1 and then go on to discuss the theory of Clifford-Fischer matrices in Section 4.2. Clifford’s Theory was developed by Clifford [8] in 1937 for ordinary representations and extended by Mackey in 1958 (see for instance Isaacs [18]) to projective representations. According to Clifford’s Theory, there is a one-to-one correspondence between the appropriate irreducible characters of ¯G and those of its subgroups called the inertia groups. Let ¯G = N·G be a group extension such that every
irreducible character of N can be extended to its inertia group. Then for each conjugacy class representative g ∈ G, a matrix M (g) called the Fischer matrix is constructed. We further discuss the theory of projective representations and characters, and establish how it is related to Clifford-Fischer Theory in Section 4.3.
In Chapter 5, we construct the character table of a maximal subgroup of M22:2, the full
automor-phism group of the Mathieu group M22, by using the method of Clifford-Fischer matrices. We
introduce the Mathieu groups and their covers in Section 5.1. In Section 5.2, we introduce the group ¯G = 24:S6, and generate it as a subgroup of M22:2. In Section 5.3, we compute the
conju-gacy classes of ¯G by using the technique of coset analysis. From the 11 conjugacy classes of S6, we
obtain 21 conjugacy classes of ¯G. In Section 5.4, we show that there are two inertia factor groups, H1 = S6 and H2 = 2 × S4. Also, we calculate the 11 Fischer matrices of ¯G by using the
Furthermore, we show how to get the full character table of ¯G by the method of Clifford-Fischer matrices. In Section 5.5, we give the full fusion map of 24:S6 into M22:2 by making use of the
following:
• the permutation character for the action of M22:2 on the cosets of 24:S6 in M22:2,
• the power maps for the conjugacy class representatives of the groups 24:S
6 and M22:2,
• the technique of set intersections of characters (discussed by Ali [1], Moori [26] and Mpono [34]).
In Chapter 6, we study the groups associated with the Higman-Sims group, HS. Seretlo in [32] and [40] discussed two group extensions associated with the Higman-Sims group and its full auto-morphism group, HS:2. These groups are, 24·S6, a maximal subgroup of HS and 25:S6, a maximal
subgroup of HS:2. Here, we discuss a maximal subgroup 25:A6 of 25:S6 in HS:2. This chapter is
organised as follows. In Section 6.1, we introduce the Higman-Sims group and its automorphism group HS:2. We then introduce the group ¯G = 25:A6, and generate it as a subgroup of HS:2 in
Section 6.2. Since 25:A6 is a maximal subgroup of 25:S6 and the fusion of 25:S6 into HS:2 was
determined by Seretlo in [40], we fuse the group ¯G = 25:A6 into 25:S6. In Section 6.3, we compute
the conjugacy classes of ¯G by using the technique of coset analysis. From the 7 conjugacy classes of A6, we obtain 24 conjugacy classes of ¯G. In Section 6.4, we show that the action of ¯G on the
irreducible characters of N , the kernel of the extension results into four orbits with corresponding inertia factor groups H1 = A6, H2 = A6, H3 = S4 and H4 = S4. In Section 6.5, we compute
the 7 Fischer matrices of ¯G by using the new approach known as the “non-combinatorial approach to Fischer matrices” (that is, we apply Programmes D and E developed by Chileshe [6] and [7]). We observe that the sizes of these matrices range from 1 to 6. We point out here that these programmes developed for Fischer matrices (Programmes D and E) only compute candidates and actual Fischer matrices with labels could be obtained from these candidates by applying the following:
• the centralizer orders of the class representatives of ¯G as computed from coset analysis cor-responding to each class representative g of G,
• if χ is a character of any group H and h is a p-singular element of H, where p is a prime, then we have χ(h) ≡ χ(hp) mod p,
• the fusion of ¯G into the main group where it sits and restrictions of characters from the main group to ¯G.
Finally in Chapter 7, we determine the conjugacy classes, Fischer matrices and character table of 26:(3·S6), a maximal subgroup of M24. The group M24 is one of the 26 sporadic simple groups
of order 244823040. The chapter is organised as follows. In Section 7.1, we introduce the group ¯
G = 26:(3·S
6), and generate it as a subgroup of M24. Further in this section, we briefly discuss the
N , the kernel of the extension and on its irreducible characters results into three orbits of lengths 1, 18 and 45 with corresponding inertia factor groups H1 = 3·S6, H2 = S5 and H3 = 2 × S4. We
further compute the conjugacy classes of ¯G by using the technique of coset analysis. We observe that, from the 16 conjugacy classes of 3·S6, we obtain 33 conjugacy classes of ¯G. In Section 7.3, we
compute the 16 Fischer matrices of ¯G by using the non-combinatorial approach to Fischer matrices. We observe that the sizes of these matrices range from 2 to 4. We further show how to obtain the full character table of ¯G by the method of Fischer matrices. In Section 7.4, we determine the fusion of 26:(3·S6) into M24.
We would like to mention that the accuracy of all the character tables constructed in this disserta-tion has been checked using Programme F in the Appendix.
2
Representation and Character Theories
In this chapter, we discuss some preliminary results on representations and characters of finite groups. In Section 2.1, we make a review of some important results in group theory needed for our further discussions. Some definitions and basic results on representations and characters are given in Section 2.2. We show how the characters of direct products of groups and factor groups can be determined in Sections 2.3 and 2.4. In Section 2.5, we consider the relationship between the characters of a group and its subgroup by methods known as restriction and induction. To achieve this, we shall give a proof of the Frobenius Reciprocity Theorem from Isaacs [18]. Finally in Section 2.6, we give some results on permutation characters that will be used in later calculations. We use as references for this chapter, Alperin [3], Isaacs [18], James [19], Pah [22], Moori [28], [29], Rotman [36] and Whitley [41]. In this chapter and the remainder of the work, we only deal with finite groups.
2.1. Some Review on Group Theory
We begin by discussing some basic concepts that will be essential for the rest of our work.
Definition 2.1.1. A subgroup H of a group G, denoted by H ≤ G, is a non-empty subset of G that forms a group under the binary operation of G.
Definition 2.1.2. If H ≤ G, then we define and denote cosets in the following way 1. Hg = {hg : h ∈ H, g ∈ G} is a set of right cosets,
2. gH = {gh : h ∈ H, g ∈ G} is a set of left cosets.
Definition 2.1.3. A subgroup H ≤ G is a normal subgroup, denoted by H E G if ∀g ∈ G, gHg−1= H.
Remark 2.1.4. If H E G, then we can define the quotient group G/H = {Hg : g ∈ G}, the set of cosets with group operations Hg1∗ Hg2 = Hg1g2, ∀g1, g2 ∈ G and H = 1G/H. The quotient
group is also refered to as, the factor group.
The set of all elements conjugate to x ∈ G is called the conjugacy class of x in G and is given by [x]G= {g−1xg : g ∈ G}. The centralizer of g ∈ G is defined by CG(g) = {h ∈ G : hg = gh}.
Proposition 2.1.6. The conjugacy classes of G form a partition of G, and an element g ∈ G has [G : CG(g)] conjugates in G.
Proof. See Proposition 13 of Alperin [3].
Definition 2.1.7. Let (G, ∗) and (H, ) be two groups. Then a homomorphism is a mapping ϕ : G −→ H such that ∀a, b ∈ G, ϕ(a ∗ b) = ϕ(a) ϕ(b). The kernel of ϕ is defined by
Ker(ϕ) = {x ∈ G : ϕ(x) = 1H}. The image of ϕ is defined by Im(ϕ) = {y ∈ H : ϕ(x) = y, x ∈ G}.
We next state and prove a result that will be prominent in the proof of Proposition 2.3.5 on the construction of all normal subgroups of a group say G.
Proposition 2.1.8. Suppose ϕ : G −→ H is a homomorphism, then Ker(ϕ) E G.
Proof. Since ϕ(1G) = 1H, then 1G ∈ Ker(ϕ) 6= ∅. For x, y ∈ Ker(ϕ), we have y−1 ∈ Ker(ϕ) since
ϕ(y−1) = (ϕ(y))−1 = 1H. Now, ϕ(xy−1) = ϕ(x)ϕ(y−1) = 1H which implies that xy−1 ∈ Ker(ϕ)
and thus, Ker(ϕ) ≤ G. Also, ∀g ∈ G and ∀x ∈ Ker(ϕ) we have,
ϕ(g x g−1) = ϕ(g) ϕ(x) ϕ(g−1) = ϕ(g) 1H (ϕ(g))−1
= ϕ(g) (ϕ(g))−1 = 1H.
Hence, g x g−1 ∈ Ker(ϕ) and therefore, Ker(ϕ) E G.
Definition 2.1.9. If M and N are groups, then their direct product, denoted by M × N , is the group with elements all ordered pairs (m, n), where m ∈ M and n ∈ N , and with operation
(m, n)(m0, n0) = (mm0, nn0).
One can easily check that M × N is a group: the identity is (1, 1) and the inverse of an element of M × N is given by (m, n)−1 = (m−1, n−1). We note that both M and N are not subgroups of M ×N , but M ×N contain isomorphic replicas of each and these are M ×{1N} = {(m, 1) : m ∈ M }
and {1M} × N = {(1, n) : n ∈ N }.
Let us now discuss the notions of group action that have the rudiments on which representations of finite groups are based. As a particular example, these notions will be illustrated in developing the theory of permutation characters in Section 2.6.
Definition 2.1.10. Let Ω = {x1, x2, · · · , xk} be a finite set. Let G be a finite group. We say that
G acts on Ω if there is a homomorphism ϕ : G −→ SΩ, where SΩ is the symmetric group on Ω
(that is SΩ is the set of all permutations of Ω ). The orbit αG where α ∈ Ω, under the action of
The following theorem asserts that under an action, a set is partitioned by the set of orbits. Theorem 2.1.11. Let the group G act on the set Ω. Then the set of all orbits of G on Ω form a partition on Ω.
Proof. Define a relation ∼ on Ω by x1 ∼ x2 if x1 = (x2)g for some g ∈ G and x1, x2 ∈ Ω. We show
that the relation ∼ is an equivalence relation. We have that x1∼ x1 since for 1G∈ G, x1= (x1)1G
and thus reflexivity holds. Suppose x1 ∼ x2, then ∃ g ∈ G such that x1 = (x2)g ⇒ x2 = (x1)g
−1
and since g−1 ∈ G, then x2 ∼ x1. Hence the relation is symmetric. If x1 ∼ x2 and x2 ∼ x3 for
x1, x2, x3∈ Ω, then for g1, g2 ∈ G, we have
x1= (x2)g1 and x2 = (x3)g2 ⇒ (x1)g
−1
1 = (x3)g2 ⇒ x1 = (x3)g2g1.
Thus, x1 ∼ x3since g2g1 ∈ G and transitivity holds. Hence the relation ∼ is an equivalence relation.
We also have that for any x ∈ Ω, [x]G= {gxg−1 : g ∈ G} = {xg : g ∈ G} = xG and therefore, the
set of all orbits of G on Ω form a partition on Ω. This completes the proof.
Definition 2.1.12. Let G be a group and Ω be a set. The stabilizer of α ∈ Ω in G is defined by Gα = {g ∈ G : αg = α}, that is, the set of all elements in G that fix α. We say that G is
transitive on Ω if αG= Ω, that is, for any α, β ∈ Ω, there exists g ∈ G such that αg = β.
The following theorem gives a link between the two concepts of orbit and stabilizer. Theorem 2.1.13 (Orbit-Stabilizer Theorem). Let G act on a finite set Ω. Then
1. Gα≤ G, for each α ∈ Ω.
2. |αG| = [G : G
α], that is, the number of elements in the orbit of α is equal to the index of Gα
in G.
Proof. 1. Since α1G= α, 1
G ∈ Gα. Hence Gα 6= ∅. If x, y ∈ Gα, then
αx= αy = α and thus, (α)xy−1 = (αx)y−1 = (αy)y−1 = α ⇒ xy−1 ∈ Gα.
Therefore, Gα≤ G as required.
2. For α ∈ Ω and g ∈ G, define ϕ : αG −→ G/Gα by ϕ(αg) = (Gα)g. We show that ϕ is a
bijection. For x, y ∈ Gα, we have αx, αy ∈ αG. Thus,
αx= αy ⇔ αxy−1 = α
⇔ xy−1∈ Gα ⇔ (Gα)xy−1 = Gα
⇔ (Gα)x = (Gα)y
⇔ ϕ(αx) = ϕ(αy).
Hence ϕ is well-defined and one-to-one. Also, ϕ is onto by definition. Thus, there is a one-to-one correspondence between αG and G/Gα and therefore, |αG| = [G : Gα].
2.2. Representations and Characters
In this section, we discuss some results from representations and characters of finite groups. There are two vital types of representations, namely, permutation and matrix representations. An example of a permutation representation is given by Cayley’s Theorem (see for instance Alperin [3] on page 28) which asserts that any group G can be embedded into the symmetric group SG. In our work,
the matrix representation of a finite group is of particular interest and we define this concept as follows:
Definition 2.2.1. Let G be a group and F be a field. Then the general linear group is a group of all n×n invertible matrices whose entries are in F and is denoted by GL(n, F). A homomorphism ϕ : G −→ GL(n, F) is called a matrix representation or simply a representation of degree n over F. If F = C, then ϕ is called an ordinary representation. A representation ϕ of G is said to be faithful if Ker(ϕ) = {1G}.
In the following, we establish how vital faithful representations are in embedding the group G in GL(n, F). By the First Isomorphism Theorem (see for instance Rotman [36] on page 35) we have that
G/Ker(ϕ) ∼= Im(ϕ).
If ϕ is a faithful representation, then Ker(ϕ) = {1G}. Therefore, G ∼= Im(ϕ) ≤ GL(n, F). Thus, the
study of Im(ϕ) is the same as the study of G.
Definition 2.2.2. Let A = (αij) be an n × n matrix. The trace of A is defined by
tr(A) =
n
X
i=1
αii.
Remark 2.2.3. 1. If A and B are two n × n matrices, then
tr(AB) = n X i=1 (AB)ii= n X i=1 n X j=1 AijBji= n X j=1 n X i=1 BjiAij = n X j=1 (BA)jj = tr(BA).
2. Similar matrices have the same trace. Suppose P and Q are two similar matrices, then there is a square matrix A such that P = AQA−1. Thus, we have that
tr(P ) = tr(AQA−1) = tr(AA−1Q) = tr(Q).
Definition 2.2.4. Let ϕ : G −→ GL(n, F) be a representation. Define χ : G −→ F by
χ(g) = tr(ϕ(g)),
then χ is called the character of G afforded by the representation ϕ. The degree of ϕ is the same as that of χ.
Definition 2.2.6. A function φ : G −→ F is said to be a class function on G if φ(gxg−1) = φ(x) for all x, g ∈ G (that is, φ is constant on each conjugacy class of G).
We now show that a character of a group is a class function in the following proposition. Proposition 2.2.7. A character χ of a group G is a class function.
Proof. Let χ : G −→ F be a character and φ be the representation corresponding to χ such that φ : G −→ GL(n, F). Then for all x, g ∈ G, we have
χ(xgx−1) = tr(φ(xgx−1))
= tr(φ(x)φ(g)(φ(x))−1) = tr(φ(g))
= χ(g).
Therefore, χ is a class function and this completes the proof.
Matrix representations can give us concrete information about finite groups and for this reason, they have to be classified. To do this, we need a notion of “equivalence” of representations. Definition 2.2.8. Two representations φ, φ0 : G −→ GL(n, F) are said to be equivalent if there exists a non-singular n × n matrix A over F such that φ(g) = A−1φ0(g)A for all g ∈ G.
In view of Definition 2.2.8, we have the following result:
Lemma 2.2.9. If φ and φ0 are equivalent representations, then χφ= χφ0.
Proof. Let φ, φ0 : G −→ GL(n, F) be two equivalent representations. Then ∃ an invertible n × n matrix A such that Aφ(g)A−1= φ0(g) for all g ∈ G. Now,
χφ0(g) = tr(φ 0
(g)) = tr(Aφ(g)A−1) = tr(φ(g)) = χφ(g).
We now establish what is meant by representations and characters being reducible, irreducible, fully-reducible or completely reducible, as these concepts will be prominent in the construction of character tables.
Definition 2.2.10. Suppose φ and ψ are representations of a group G such that ∀ g ∈ G,
ψ(g) = A1(g) A2(g) 0 A3(g)
! ,
then φ is reducible if it is equivalent to ψ. If φ is not reducible, then it is said to be irreducible. Now ∀ g ∈ G, define ψ by
ψ(g) = B1(g) 0 0 B2(g)
! ,
we say that φ is fully reducible if it is equivalent to ψ. φ is said to be completely reducible if ∃ P ∈ GL(n, F) such that ∀ g ∈ G, P−1φ(g)P = C1(g) 0 · · · 0 0 C2(g) · · · 0 .. . ... . .. ... 0 0 · · · Ck(g)
where the Ci’s (i ∈ {1, 2, · · · , k}) are irreducible.
Definition 2.2.11. The character χφ afforded by an irreducible representation φ is called an
ir-reducible character.
We next give a way of defining new representations and characters from known ones. This further sheds more light on irreducibility of representatios and characters.
Definition 2.2.12. Let φ : G −→ GL(n, F) and ψ : G −→ GL(m, F) be two representations (n = m or n 6= m). Then ∀ g ∈ G, the sum of the representations φ ⊕ ψ : G −→ GL(n + m, F) is defined by
(φ ⊕ ψ)(g) = φ(g) 0 0 ψ(g)
! .
Proposition 2.2.13. The sum of representations is a representation.
Proof. If φ and ψ are representations of a group G given in Definition 2.2.12, then we show that φ ⊕ ψ : G −→ GL(n + m, F) is a homomorphism. Thus, for all g, h ∈ G we have
(φ ⊕ ψ)(gh) = φ(gh) 0 0 ψ(gh) ! = φ(g)φ(h) 0 0 ψ(g)ψ(h) ! = φ(g) 0 0 ψ(g) ! . φ(h) 0 0 ψ(h) ! = (φ ⊕ ψ)(g).(φ ⊕ ψ)(h)
and therefore, φ ⊕ ψ is a representation.
Remark 2.2.14. A representation of a group is said to be irreducible if it cannot be expressed as a sum of other representations. From Definition 2.2.12, χφ+ χψ is the character of φ ⊕ ψ. We also
note that an irreducible character of a group cannot be expressed as a sum of other characters. We will denote the set of all irreducible characters of a group G by Irr(G).
In the following, we discuss Maschke’s Theorem which is well known in determining irreducible representations from which we obtain the irreducible characters as stated in Definition 2.2.11 and Remark 2.2.14.
Theorem 2.2.15 (Maschke’s Theorem). Let G be a finite group and φ : G −→ GL(n, F) be a representation over the field F of characteristic either zero or a prime p which does not divide |G|. If φ is reducible, then it is fully reducible.
Proof. See Theorem 8.1 of James [19].
Considering the condition of Maschke’s Theorem, the representation φ is completely reducible i.e, it can be expressed as a sum of irreducible representations of G in the following manner. If φ is irreducible then obviously it is completely reducible by Definition 2.2.10. If φ is reducible, then it is fully reducible by Maschke’s Theorem. Thus, for all g ∈ G, ∃ an invertible matrix P such that
P−1φ(g)P = A(g) 0 0 B(g)
! .
We note that A and B are the following representations; A : G −→ GL(r, F), B : G −→ GL(s, F), where r + s = n and φ = A ⊕ B. Since A and B are representations of G over F, we can repeatedly apply Maschke’s Theorem to them and obtain
φ(g) = B1(g) 0 · · · 0 0 B2(g) · · · 0 .. . ... . .. ... 0 0 · · · Bk(g)
where, the Bi’s are all irreducible. Hence, φ = B1⊕ B2⊕ · · · ⊕ Bk.
In the next section, we discuss the inner product and orthogonality relations of characters of a group. These two concepts are essential in completing the character tables of finite groups.
2.2.1 Inner product and orthogonality relations
Definition 2.2.16. Let G be a group and F be a field such that the characteristic of the field denoted by char(F), does not divide |G|. If φ and ψ are two class functions from G into F, then we define the inner product of φ and ψ by
hφ, ψi = 1 |G|
X
g∈G
φ(g)ψ(g−1).
In a similar manner, if χ1and χ2are two characters of the group G, then hχ1, χ2i =
1 |G|
X
g∈G
χ1(g)χ2(g−1)
since characters are class functions by Proposition 2.2.7.
From now on, we will consider representations and characters of a finite group G over the complex field C.
Remark 2.2.17. Suppose φ : G −→ GL(n, C) is a representation of a group G, then for g ∈ G, we denote the (i, j) entry of φ(g) by φij(g). Thus we can regard φij to be a map from G into C.
We next explore the concept of inner product and how it can be used to determine irreducibility of characters of a group. Theorem 2.2.18 and its proof can be found in Moori [29].
Theorem 2.2.18. Let G be a group. Let φ and ψ be irreducible representations of G such that χφ
and χψ are characters afforded by φ and ψ respectively.
1. If φ and ψ are inequivalent representations, then for all i, j, k and l, hφij, ψkli = 0.
2. hχφ, χφi = 1.
3. hχφ, χψi =
1 if φ and ψ are equivalent, 0 otherwise.
Proof. 1. See Theorem 5.2.1 (i) of Moori [29].
2. Let the degree of φ be r. Then for all g ∈ G, we have
hχφ, χφi = 1 |G| X g∈G χφ(g)χφ(g) = 1 |G| X g∈G χφ(g)χφ(g−1) = 1 |G| X g∈G {[ r X i=1 φii(g)][ r X j=1 φjj(g−1)]} = r X i=1 r X j=1 [ 1 |G| X g∈G φii(g)φjj(g−1)] = r X i=1 r X j=1 hφii, φjji = r X i=1 1 r (since r X j=1 hφii, φjji = 1 r) = r.1 r = 1. 3. Follows from parts 1 and 2.
We recall from Definition 2.2.11 that χφ is an irreducible character since it is afforded by an
irreducible representation φ. Thus, from Theorem 2.2.18 (2), we note that the inner product of an irreducible character χφis 1.
The following lemma, asserts that the study of inequivalent irreducible representations of G pro-duces the distinct irreducible characters of G which are well understood in the character table of
G.
Lemma 2.2.19. Two irreducible representations ζ1 and ζ2 of a group G are equivalent if and only
if χζ1 = χζ2.
Proof. If ζ1 and ζ2 are equivalent irreducible representations of a group G, then by Lemma 2.2.9,
χζ1 = χζ2. Conversely, suppose χζ1 = χζ2, then we have hχζ1, χζ2i = 1 by Theorem 2.2.18 (2). Hence
by Theorem 2.2.18 (3), ζ1 and ζ2 are equivalent and irreducible. This completes the proof.
Remark 2.2.20. From the above lemma, if ζ1 and ζ2 are inequivalent irreducible representations,
then χζ1 and χζ2 are two distinct irreducible characters (i.e, χζ1 6= χζ2).
It was pointed out in Proposition 2.2.7 that the characters of a group G are class functions on G. Since the class functions are constant on the conjugacy classes by Definition 2.2.6, then there is a connection between the conjugacy classes of a group G and the irreducible characters. This connection is cardinal in the construction of character tables. The following theorem gives that connection.
Theorem 2.2.21. The number of irreducible characters of a group G is equal to the number of conjugacy classes of G.
Proof. See Theorem 15.3 of James [19].
We now discuss orthogonality relations which often make it possible to complete the character table of a group when some of the irreducible characters are known.
Theorem 2.2.22. If G is a group and Irr(G) = {χ1, χ2, · · · , χr}, then the following holds for any
s ∈ G. 1 |G| X g∈G χi(gs)χj(g−1) = δij χi(s) χi(1G) .
Proof. See Theorem 2.13 of Isaacs [18].
Theorem 2.2.23. (Alperin [3]) Let G be a group and Irr(G) = {χ1, χ2, · · · , χr}. If {g1, g2, · · · , gr}
are the set of conjugacy class representatives of G, such that {κ1, κ2, · · · , κr} are the orders of the
conjugacy classes, then for any 1 ≤ i, j ≤ r 1. we have the row orthogonality relation
hχi, χji = 1 |G| X g∈G χi(g)χj(g) = 1 |G| r X t=1 κtχi(gt)χj(gt) = δij,
2. we have the column orthogonality relation
r X t=1 χt(gi)χt(gj) = δij |G| κi = δij|CG(gi)|.
2. Let X = (χi(gj)) be the character table of the group G. Let M be the diagonal matrix with
entries {κ1, κ2, · · · , κr}. Then for any i and j we have,
XM (X)T =
r
X
t=1
χi(gt)κt(X)T,
where (X)T is the transpose of X and XM (X)T is an i × j invertible matrix. Thus, by the row orthogonality relation we obtain
XM (X)T = r X t=1 κtχi(gt)χj(gt) = X g∈G χi(g)χj(g) = |G|hχi, χji = |G|δij.
Hence, we have a system of r2 equations given by XM (X)T = |G|I, where I is an r × r identity matrix. Since XM (X)T is a non-zero matrix, then |G|I = M X(X)T so that
|G|δij = |G|I = r X t=1 (M XT)jtXti= κi r X t=1 χt(gj)χt(gi). (2.1)
Dividing equation 2.1 by κi we obtain r X t=1 χt(gi)χt(gj) = δij |G| κi = δij|CG(gi)| as required.
Remark 2.2.24. If i = j in the column orthogonality relation 2, then gi = gj so that r X t=1 |χt(gi)|2= |CG(gi)| otherwise r X t=1
χt(gi)χt(gj) = 0. This implies that r
X
t=1
|χt(1G)|2 = |G|, which
is a prominent relation in the determination of the degrees of the Irr(G).
Having discussed the inner product and orthogonality relations, we proceed to discuss the character table of a group G in the next subsection.
2.2.2 Character tables
The character table of a group contains important information to study any finite group G. By Theorem 2.2.21, the number of irreducible characters of a group G is equal to the number of conjugacy classes of G. Thus the character table is viewed as a square matrix with rows indexed by irreducible characters and columns corresponding to the conjugacy class representatives of G.
Definition 2.2.25. Let {g1, g2, · · · , gr} be conjugacy class representatives of G and {χ1, χ2, · · · , χr}
be a set of irreducible characters of G. Then the r × r matrix (say X) whose entries are the values χi(gj) for i, j ∈ {1, 2, · · · , r} is called the character table of G.
For the character table X, we usually take χ1 as the trivial character and set g1 = 1G, so that the
first column of the character table consists of degrees of the Irr(G) discussed in Remark 2.2.24. We will generally write X in the form of Table 2.1 where the di’s are the degrees of the Irr(G), and
ki= CG(gr) = [G : |[gr]G|] for i ∈ {2, 3, · · · , r}.
Table 2.1: General Form of a Character Table
[gr]G 1G g2 · · · gr CG(gr) |G| k2 · · · kr χ1 1 1 · · · 1 χ2 d2 χ2(g2) · · · χ2(gr) .. . ... ... . .. ... χr dr χr(g2) · · · χr(gr)
As alluded to earlier on, certain properties of the group G can be deduced from its character table. For instance, we can
1. determine the order of G as given by Remark 2.2.24,
2. establish whether G is simple or not, (i.e, for any non-trivial element gr ∈ G and any
non-trivial irreducible character χr of G, if χr(gr) 6= χr(1G), then the group G is simple otherwise
it is not simple,
3. determine whether G is abelian that is, if and only if all its irreducible characters are linear, 4. determine all the normal subgroups of G. We shall further discuss this fact in Section 2.3.
2.3. Normal Subgroups and Lifted Characters
This brief section presents a method for constructing characters of a group G and later discusses the construction of all normal subgroups of G. If N is a proper normal subgroup of G, then we may look at the factor group G/N to be smaller than G. Thus, it should be easier to construct the characters of G/N than those of G. In fact, we may use the characters of G/N to construct some of the characters of G by a process which is known as lifting. Therefore, the normal subgroups help us to find the characters of G and on the other hand the character table of G enables us to determine all the normal subgroups of G.
2.3.1 Lifted characters
In the following, we show how a character of G can be constructed from a character of a factor group G/N .
Proposition 2.3.1. Let N C G and χ be a character of G/N . Then ∀g ∈ G, the functione
χ : G −→ C defined by χ(g) =χ(N g) is a character of G with deg(χ) = deg(e χ).e
Proof. Let ϕ : G/N −→ GL(n, C) be a representation which affords a charactere χ ande
ϕ : G/N −→ GL(n, C) be a function defined by ϕ(g) =ϕ(N g), ∀g ∈ G. Then ∀g, h ∈ G we have thate
ϕ(gh) = ϕ(N gh)e = ϕ(N gN h)e = ϕ(N g)e ϕ(N h)e = ϕ(g)ϕ(h)
and therefore, ϕ defines a representation on G. Thus, if χ is the character afforded by ϕ then ∀g ∈ G,
χ(g) = tr(ϕ(g)) = tr(ϕ(N g)) =e χ(N g)e so that χ is the character of G. Finally, for the degree of χ and χ we havee
deg(χ) = χ(1G) =χ(N 1e G) =χ(N ) = deg(e χ).e
This completes the proof.
Definition 2.3.2. Let χ and χ be as defined in Proposition 2.3.1. Then the character χ is callede the lift of χ to G.e
Remark 2.3.3. From Proposition 2.3.1 we note that, if χ ∈ Irr(G/N ), then χ ∈ Irr(G). Thise fact can be argued in the following way. If χ ∈ Irr(G/N ), then by Theorem 2.2.18 (2), he χ,e χi = 1.e Suppose we take T to be a transversal of N in G, then
hχ,e χie = 1 |G/N | X N g∈G/N e χ(N g)χ(N g)e −1 = |N | |G| X N g∈G/N e χ(N g)χ(N g)e −1 = 1 |G| X g∈T |N |χ(N g)e χ(N ge −1) = 1 |G| X g∈T |N |χ(g)χ(g−1) = 1 |G| X g∈G χ(g)χ(g−1) = hχ, χi = 1.
Therefore, χ is an irreducible character of G.
With the results obtained so far, we now go on to state a theorem which enables us to write down as many irreducible characters of G as there are irreducible characters of G/N .
Theorem 2.3.4. Let N C G. Then there is a bijective correspondence between the set of characters e
χ of G/N and the set of characters χ of G which satisfy N ≤ Ker(χ). Irreducible characters of G/N correspond to the irreducible characters of G which have N in their kernel.
Proof. See Theorem 17.3 of James [19].
The results of Theorem 2.3.4 will play a big role in calculating Clifford-Fischer matrices using the new approach to be discussed in Chapter 4.
2.3.2 Normal subgroups
In this subsection, we show how to construct all the normal subgroups of a group G. From Definition 2.1.7, we can deduce that Ker(χ) = {g ∈ G : χ(g) = χ(1G)} and therefore, we can easily locate
the kernel of an irreducible character χ from the character table of G. Also, by Proposition 2.1.8, Ker(χ) E G. Thus, the following Proposition 2.3.5 asserts that every subgroup of G which is the intersection of the kernels of some irreducible characters, is a normal subgroup. Hence all normal subgroups of G can be constructed in this manner.
Proposition 2.3.5. Let N be a proper normal subgroup of G. Then there exits Irr(G) = {χ1, χ2, · · · , χr}
such that N = r \ i=1 Ker(χi).
Proof. Let Irr(G) = {χ1, χ2, · · · , χm}. Then by the definition of a kernel, m
\
i=1
Ker(χi) = {1G}. If
G/N has r distinct irreducible characters χe1,χe2, · · · ,χer, then
r
\
i=1
Ker(χei) = {N } (since N is an
identity element of G/N ). For 1 ≤ i ≤ r, let χi be the lifts to G of χei. Thus, if g ∈ Ker(χi), then
e
χi(N ) = χi(1G) = χi(g) =χei(N g),
and hence N g ∈ Kerχei. Therefore, if g ∈ |Irr(G)| \ i=1 Ker(χi), then N g ∈ r \ i=1
Ker(χei) = {N }, and thus
g ∈ N . Since the choice of g was arbitrary, it follows that N =
r
\
i=1
Ker(χi).
2.4. Products of Characters
In Section 2.2, we referred to ways of obtaining new representations from the known ones, we now explore the question of independently obtaining new characters from the known characters and also finding all the irreducible characters of a direct product M × N , given those of M and N . Most of the development of this follows Moori [29] and Whitley [41].
Definition 2.4.1. If A = (aij)m×m and B = (bij)n×n are two matrices, then the product given by
A ⊗ B = a11B a12B · · · a1mB a21B a22B · · · a2mB .. . ... . .. ... am1B am2B · · · ammB ,
is called the tensor product of A by B and is an mn × mn matrix. Remark 2.4.2. From Definition 2.4.1, we have the following results:
1. The trace of A ⊗ B is given by
tr(A ⊗ B) = a11tr(B) + a22tr(B) + · · · + ammtr(B)
= (a11+ a22+ · · · + amm)tr(B)
= tr(A)tr(B).
2. If A0 = (a0ij)r×r and B0 = (b0ij)s×s are matrices, then
(A ⊗ B)(A0⊗ B0) = AA0⊗ BB0.
In the following, we define the tensor product of two representations.
Definition 2.4.3. Let φ and ψ be two representations of G. We define the tensor product φ ⊗ ψ, by
We now have the following Proposition 2.4.4 motivated by Definitions 2.4.1 and 2.4.3.
Proposition 2.4.4. Let φ : G −→ GL(m, F) and ψ : G −→ GL(n, F) be representations. Then 1. φ ⊗ ψ : G −→ GL(mn, F) forms a representation.
2. χφ⊗ψ = χφχψ.
Proof. 1. We show that ∀g, h ∈ G, φ ⊗ ψ : G −→ GL(mn, F) is a homomorphism. Now, (φ ⊗ ψ)(gh) = φ(gh) ⊗ ψ(gh), by Definition 2.4.3 = φ(g)φ(h) ⊗ ψ(g)ψ(h), = (φ(g) ⊗ ψ(g))(φ(h) ⊗ ψ(h)), by Remark 2.4.2 (2) = (φ ⊗ ψ)(g)(φ ⊗ ψ)(h), by Definition 2.4.3. 2. ∀g ∈ G, χφ⊗ψ(g) = tr((φ ⊗ ψ)(g)), = tr(φ(g) ⊗ ψ(g)), by Definition 2.4.3 = tr(φ(g))tr(ψ(g)), by Remark 2.4.2 (1) = χφ(g)χψ(g), = (χφχψ)(g).
Proposition 2.4.4 justifies that products of characters afforded by products of representations are also characters of G. We also note that the tensor product of characters is commutative
(that is, χφ⊗ψ(g) = χφ(g)χψ(g) = χψ(g)χφ(g) = χψ⊗φ(g) ) but in general, (φ ⊗ ψ)(g) 6= (ψ ⊗ φ)(g).
The following theorem asserts that if the irreducible characters of two groups M and N are known, then the tensor product of these groups can be used to obtain the irreducible characters of their direct product M × N . We discussed what a direct product of groups is in Definition 2.1.9.
Theorem 2.4.5. Let M and N be two groups with conjugacy classes C1, C2, · · · , Csand C10, C20, · · · , Ct0
respectively. Suppose that Irr(M ) = χ1, χ2, · · · , χs and Irr(N ) = χ01, χ02, · · · , χ0t. The conjugacy
classes of M × N are Ci× Cj0 and Irr(M × N ) = {χi× χ0j : χi ∈ Irr(M ), χ0j ∈ Irr(N )} for 1 ≤ i ≤ s
and 1 ≤ j ≤ t.
Proof. See Theorem 5.3.2 of Moori [29].
Remark 2.4.6. It is clear from Theorem 2.4.5 that the character table of any direct product M ×N can be constructed from the character tables of M and N .
It is important to note that the product of two irreducible characters is not in general an irreducible character. The following proposition sheds more light.
Proposition 2.4.7. Let χ be a linear character of G and χ0∈ Irr(G). Then χχ0 ∈ Irr(G).
Proof. Let χ be a linear character of G (that is, deg(χ) = 1). Then ∀ g ∈ G, χ(g) is the root of unity. In particular, for any g ∈ G, we have 1 = |χ(g)|2= χ(g)χ(g). If χ0 ∈ Irr(G), then
hχχ0, χχ0i = 1 |G| X g∈G χχ0(g)χχ0(g) = 1 |G| X g∈G χ(g)χ0(g)χ(g) χ0(g) = 1 |G| X g∈G χ(g)χ(g)χ0(g)χ0(g) = 1 |G| X g∈G χ0(g)χ0(g) = hχ0, χ0i = 1.
Therefore, χχ0 is an irreducible character of G.
2.5. Restriction and Induction of Characters
We now discuss how to obtain representations and characters of subgroups from the main group and vice versa. These two dual operations are known as restriction and induction of characters. This section is critical in understanding Clifford’s Theory and Clifford-Fischer matrices.
2.5.1 Restriction to a subgroup
If the representations of a group say G are known, we aim to study how we can obtain the repre-sentations and in like manner, the characters of a subgroup H of G.
Definition 2.5.1. Let H ≤ G and let ψ : G −→ GL(n, F) be a representation of G. The restric-tion of ψ to H is defined by
ψ↓GH(h) = ψ(h), ∀h ∈ H. In the same manner, the restriction of χψ to H is defined by
χψ↓GH(h) = χψ(h), ∀h ∈ H.
Remark 2.5.2. If χψ is an irreducible character of G, then it does not necessarily mean that χψ↓GH
is an irreducible character of H.
We now explore some of the important properties of restricted characters. The next proposition shows that every irreducible character of H is a constituent of the restriction of some irreducible character of G.
Proposition 2.5.3. Let φ be a character of H ≤ G . Then there exists an irreducible character χ of G such that
hχ↓G
H, φi 6= 0.
Proof. See Proposition 20.4 of James [19].
Theorem 2.5.4. Let H ≤ G. Let χ ∈ Irr(G) and Irr(H) = {φ1, φ2, · · · , φk} be the irreducible
characters of H. Then χ↓GH =
k
X
i=1
eiφi, where the e,is are non-negative integers and
k X i=1 e2i ≤ |G : H|. Moreover, k X i=1
e2i = |G : H| if and only if χ(g) = 0 for all g ∈ G \ H.
Proof. See Theorem 5.4.2 of Moori [29].
Remark 2.5.5. Theorem 2.5.4 asserts that the number of irreducible constituents of ψ↓GH is bounded above by [G : H]. Therefore, if [G : H] is fairly small, the character tables of H and G are closely related.
Next we discuss some results of obtaining the characters of the main group knowing those of its subgroup.
2.5.2 Induction to the main group
Let H ≤ G. If ϑ is a character of H, then we can obtain a character of G from ϑ.
Definition 2.5.6. Let H ≤ G and ϑ be a character of H. Then ϑ↑GH, the induced character of ϑ to G, is defined by ϑ↑GH = 1 |H| X x∈G ϑo(xgx−1), ∀g ∈ G
where ϑo is an extension of ϑ to G defined by
ϑo(h) = ϑ(h) if h ∈ H, 0 if h /∈ H.
We note that deg(ϑ↑GH) = deg(ϑ)[G : H].
Proposition 2.5.7 below is vital in the evaluation of the values of induced character ϑ↑GH on classes of G.
Proposition 2.5.7. Let H ≤ G and [g]G be a conjugacy class of G. Let {x1, x2, · · · , xk} be a set
of representatives of the conjugacy classes of H that fuse to g. Let ϑ be a character of H. If
H ∩ [g]G= ∅, then ϑ↑G H(g) = 0,
non-empty set, then ϑ↑GH(g) = |CG(g)| k X i=1 ϑ(xi) |CH(xi)| .
Proof. Let g be a conjugacy class representative of G. If H ∩ [g]G = ∅, then ∀ x ∈ G, xgx−1 ∈ H/
and by Definition 2.5.6, ϑo(xgx−1) = 0 for any x ∈ G so that ϑ↑GH(g) = 0. Now if H ∩ [g]G 6= ∅,
then let h ∈ H ∩ [g]G. As x ranges over G, we have h = xgx−1 covering [g]G exactly |CG(g)| times,
so ϑ↑GH(g) = |CG(g)|
|H|
X
t∈[g]G
ϑo(t). If t /∈ H, then we have ϑo(t) = 0, and H ∩ [g]
Gcontains [H : CH(xi)]
conjugates of each xi. Therefore, ϑ↑GH(g) = |CG(g)| k X i=1 ϑ(xi) |CH(xi)| as required.
Next, we discuss a result of a German mathematician, Ferdinand Georg Frobenius, which gives the relation between restriction and induction of characters, referred to as the Frobenius reciprocity. Theorem 2.5.8. (Isaacs [18])(Frobenius Reciprocity Theorem). Let H ≤ G. Let ϑ and χ be characters of H and G respectively. Then the following holds:
hϑ, χ↓GHiH = hϑ↑GH, χiG. Proof. By Definition 2.2.16, hϑ↑GH, χiG = 1 |G| X g∈G ϑ↑GH(g)χ(g), and by Definition 2.5.6, hϑ↑G H, χiG = 1 |G| 1 |H| X g∈G X x∈G ϑo(xgx−1)χ(g). If h = xgx−1, then hϑ↑GH, χiG = 1 |G| 1 |H| X h∈G X x∈G ϑo(h)χ(h) = 1 |H| X h∈G ϑ(h)χ(h) = hϑ, χ↓GHiH.
Remark 2.5.9. Suppose that ϑ ∈ Irr(H) and χ ∈ Irr(G) where H ≤ G. If ϑ↑GH =
p X i=1 dijχi and χ↓GH = q X j=1
eijϑj, then by Frobenius Reciprocity Theorem, we get dij = hϑ↑GH, χii = hχ↓GH, ϑji = eij.
As we conclude this section, it is important to note that, like restriction (see Remark 2.5.2), induction of characters do not necessarily preserve irreducibility of characters.
2.6. Permutation Character
To develop the theory on permutation characters, we shall use some notions from group action discussed in Section 2.1. In this section, we discuss the main result that any subgroup H of G will give rise to a permutation character 1↑GH where G acts transitively on the cosets of H in G. If the group G acts transitively on the finite set Ω, then χ(G|Gα) = 1↑GGα, α ∈ Ω is the resulting
permutation character. This result will play an important role in determining the number of fixed points by the action of G on the cosets of H in G. We now define the permutation representation and character as follows:
Definition 2.6.1. Let G act on a finite set Ω = {x1, x2, · · · , xr} and for each g ∈ G, define the
r × r matrix πg by πg = (βij), where βij = 1 if xgi = xj, 0 otherwise.
Then under the action of g, πg is a permutation matrix and ψ : G −→ GL(r, C) given by
ψ(g) = πg, is a permutation representation of G. The character χψ afforded by the
represen-tation ψ is called a permurepresen-tation character.
From the above definition, it is clear that the permutation character,
χψ(g) = tr(ψ(g)) = χπ(g) = tr((βij)) = |{x ∈ Ω : xg= x}|,
that is, χψ(g) is the number of points of Ω fixed by g ∈ G. Therefore,
deg(χψ(g)) = χψ(1G) = |{x ∈ Ω : x1G= x}| = |Ω|,
and χψ(g) is always a non-negative integer.
Remark 2.6.2. Let H be a subgroup of G and T = {x1, x2, · · · , xk} be a right transversal of H in
G. Then for all g ∈ G, G acts on G/H by (Hxi)g = Hxig. Clearly, this action is transitive since
for any xi, xj ∈ T , we have (Hxi)x
−1
i xj = Hxix−1
i xj = Hxj. The transitive action gives rise to a
permutation character which is in fact a trivial character of H induced to G (that is, the character 1↑GH). To see this, we note that for any g ∈ G,
(Hxi)g = Hxi⇐⇒ Hxig = Hxi⇐⇒ Hxigx−1i = H ⇐⇒ xigx−1i ∈ H. (2.2)
Using Definition 2.5.6, denote by χ(G|H), the permutation character of G on the cosets of H in G, then χ(G|H)(g) =
k
X
i=1
πo(xigx−1i ), where πo(t) = π(t) if t ∈ H and 0 if t /∈ H. But if
t ∈ H, then by Definition 2.6.1, π(t) = 1. Since xigx−1i ∈ H by relation (2.2), it then follows that
χ(G|H)(g) = k X i=1 π(xigx−1i ) = k X i=1 1 = k = [G : H] and therefore χ(G|H)(g) = 1↑GH(g).
Remark 2.6.2 intels that for any subgroup H, there is always a permutation character of G resulting from its action on the cosets of H in G. The converse of the above remark is also true and this is the assertion of the following Theorem 2.6.3.
Theorem 2.6.3. Let G be a group acting transitively on Ω. Let α ∈ Ω, H = Gα and χ(G|H) be
the permutation character of this action. Then
χ(G|H) = 1↑GGα.
Proof. The idea of this proof follows as in Whitley [41]. Since G acts transitively on Ω, we have αG = Ω. Thus by the Orbit-Stabilizer Theorem, there is a one-to-one correspondence between Ω and the set of right cosets of the stabilizer Gα in G, given by αy −→ (Gα)y for y ∈ G. Now if
g ∈ G, then (αy)g = αy ⇐⇒ (αy)gy−1 = α ⇐⇒ ygy−1∈ Gα ⇐⇒ (Gα)ygy−1 = Gα ⇐⇒ (Gα)yg = (Gα)y ⇐⇒ [(Gα)y]g = (Gα)y,
where, G acts on the right cosets of Gα in G as given in Remark 2.6.2. Therefore, the permutation
character of the action of G on Ω is the same as the permutation character of the action of G on the right cosets of Gα in G given by 1↑GGα.
Corollary 2.6.4. (Isaacs [18]) Let G act on the set Ω with χ(G|Ω) as the permutation character of the action. Suppose Ω splits into exactly k orbits under the action of G. Then for the trivial character 1 of G we have
hχ(G|Ω), 1i = k.
Proof. Let {A1, A2, · · · , Ak} be the orbits of Ω under the action of G. Then Ω = k
[
i=1
Ai. Let Gα be
the stabilizer of α ∈ Ai and let χi(G|Gα) be the permutation character of G on the cosets of Gα.
Then we obtain χ(G|Ω) = k X i=1 χi(G|Gα) (2.3)
where, χi(G|Gα) = 1↑GGα by Theorem 2.6.3. Now by the Frobenius Reciprocity Theorem, we have
that
hχi(G|Gα), 1iG= h1↑GGα, 1iG= h1, 1↓
G
GαiGα = 1. (2.4)
Therefore, combining results of equations (2.3) and (2.4), we obtain
hχ(G|Ω), 1)i = k X i=1 hχi(G|Gα), 1i = k X i=1 1 = k
From Corollary 2.6.4, we observe that G acts transitively on Ω if and only if hχ(G|Ω), 1)i = 1. We next discuss Lemma 2.6.5 to help us understand some results of Theorem 2.6.6.
Lemma 2.6.5. Let G act transitively on Ω. Then all subgroups G%, % ∈ Ω of G are conjugate in
G.
Proof. For all %, σ ∈ Ω, we show that G% and Gσ are conjugate in G. Now, there is some h ∈ G
such that %h= σ, that is, % = σh−1 since G is transitive on Ω. Thus, g ∈ G%⇐⇒ %g = %
⇐⇒ σh−1g = σh−1 ⇐⇒ σh−1gh= σ
⇐⇒ h−1gh ∈ Gσ
⇐⇒ g ∈ (Gσ)h.
Hence, G% = (Gσ)h, which shows that G% = hGσh−1. Therefore, G% and Gσ are conjugate in
G.
We have already seen that any subgroup H of G gives rise to a permutation character and the converse also holds if we can identify the character 1↑GH. Since this character is a transitive per-mutation character by Theorem 2.6.3, it must satisfy the following conditions which we collect in Theorem 2.6.6.
Theorem 2.6.6. (Whitley [41]) Let H ≤ G and χ(G|H) = 1↑GH be the permutation character by the action of G on H. Then
1. deg(χ(G|H)) divides |G|.
2. χ(G|H)(g) is a non-negative integer, ∀g ∈ G. 3. hχ(G|H), 1Gi = 1.
4. hχ(G|H), φi ≤ deg(φ) for any φ ∈ Irr(H). 5. χ(G|H)(g) = 0 if o(g) - |G|
χ(G|H)(1).
6. χ(G|H)(g) ≤ χ(G|H)(gk) for all g ∈ G and k a non-negative integer. 7. χ(G|H)(g) |[g]G|
χ(G|H)(1G) ∈ Z for all g ∈ G.
Proof. If Ω is a set of right cosets of H ∈ G, then it follows that χ(G|H) is the permutation character of G on Ω.
1. Since deg(χ(G|H)) = [G : H] by Remark 2.6.2, then |G|
deg(χ(G|H)) = |H| and therefore, deg(χ(G|H)) divides |G|.
2. This is clear since χ(G|H)(g) is the number of points of Ω fixed by g and therefore, must be non-negative integer.
3. Since χ(G|H) is a transitive permutation character, it follows by corollary 2.6.4 that χ(G|H) contains the trivial character of G once (that is, hχ(G|H), 1Gi = 1).
4. Using Frobenius Reciprocity Theorem, we have that
hχ(G|H), φiG= h1↑GH, φiG= h1↓GH, φ↓GHiG≤ deg(φ).
5. From (a) above, we have that |H| = |G|
χ(G|H)(1) and so if o(g) does not divide H, then H ∩ [g]G= ∅ and by Proposition 2.5.7, χ(G|H)(g) = 1↑GH = 0.
6. If g ∈ Gα, α ∈ Ω, then αg = α and for any non-negative integer k, (αg)k = α. Hence any
point of Ω fixed by g is also fixed by gk. So the number of points fixed by g does not exceed those fixed by gk and therefore, χ(G|H)(g) ≤ χ(G|H)(gk).
7. Let $ = {(σ, y) : σ ∈ Ω, y ∈ [g]G, σy = σ}. Since by Definition 2.2.6, χ(G|H) is constant on
[g]G , then
χ(G|H)|[g]G| = |$| =
X
σ∈Ω
|Gσ∩ [g]G|.
Now by Lemma 2.6.5, all subgroups Gσ are conjugate in G. Therefore, |Gσ∩ [g]G| = k is
independent of σ, and
χ(G|H)|[g]G| = k
X
σ∈Ω
1 = k|Ω| = kχ(G|H)(1G).
This completes the proof.
Remark 2.6.7. Using Proposition 2.5.7, we note that if
H ∩ [g]G= ∅, then 1↑G H(g) = 0 non-empty, then 1↑GH(g) = k X i=1 |CG(g)| |CH(xi)| ,
where {x1, x2, · · · , xk} is the set of class representatives of H that fuse to g.
3
Group Extensions
In this chapter, we discuss the general theory of group extensions. In Section 3.1, we introduce notations, give definitions and present basic results on group extensions. In Section 3.2, we discuss a result from Scott [39] that every split extension ¯G of N by G, where N is a normal subgroup of
¯
G, is identical to a semi-direct product of N by G. We also show that every semi-direct product ¯
G of N by G realizes a homomorphism θ : G −→ Aut(N ). Finally in Section 3.3 we discuss the technique of coset analysis for computing conjugacy classes of group extensions. Coset analysis is applicable to both split and non-split extensions. This technique was developed by Moori in [26], [27] and has since been widely used for computing the conjugacy classes of several group extensions in all cases where it is applicable. For instance, it has been used in Ali [1] and [2], Basheer and Moori [4], Chileshe, Moori and Seretlo [6] and [7], Mpono [34], Rodriques [35], Salleh [37] and Whitley [41]. We mainly use as our references in this chapter, Chileshe [6], Moori [26], [27], Mpono [34], Rotman [36], Scott [39] and Whitley [41].
3.1. Definitions and Basic Results
Suppose that {· · · , Mn−1, Mn, Mn+1, · · · } and {· · · , λn−1, λn, λn+1, · · · } are sets of groups and
ho-momorphisms respectively. Then we call
· · ·λ−→ Mn−1 n−1−→ Mλn n λn+1
−→ Mn+1−→ · · · (3.1)
a sequence of groups and homomorphisms. The sequence (3.1) is said to be exact if for each successive pair (λn, λn+1), we have that Im(λn) = Ker(λn+1).
We now define a group extension as follows
Definition 3.1.1. Let { ¯G, G, N } and {α, β} be sets of groups and homomorphisms respectively. Then a short exact sequence is an exact sequence of the form
{1} −→ N −→ ¯α G−→ G −→ {1}.β (3.2) Given a short exact sequence (3.2), we call ¯G containing N an extension of N by G and denote it by ¯G = N·G.
In the next proposition, we state and prove some results that will be significant in the discussion of semi-direct products, split and non-split extensions.
Proposition 3.1.2. If ¯G is an extension of N by G given by the short exact sequence
{1} α1
−→ N α2
−→ ¯G−→ Gα3 α4
−→ {1}, (3.3)
then the following holds: 1. α2(N ) ∼= N ,
2. N E ¯G, 3. ¯G/N ∼= G.
Proof. Since ¯G is an extension of N by G given by a short exact sequence (3.3), then for 1 ≤ i ≤ 4, the αi’s are all homomorphisms and Im(αi) = Ker(αi+1) for any successive pair (αi, αi+1).
1. We have that α1({1}) = Im(α1) = {1} = Ker(α2). By the First Isomorphism Theorem,
N = N/Ker(α2) ∼= Im(α2) = α2(N ) and the result follows.
2. Since N ≤ ¯G and N ∼= Im(α2) = Ker(α3) E ¯G (see Proposition 2.1.8), then it follows that
N E ¯G and the result holds.
3. From the sequence (3.3), α4 : G −→ {1} implies that Im(α3) = Ker(α4) = G. Now by 1, 2
and the First Isomorphism Theorem we obtain ¯
G/N ∼= ¯G/α2(N ) = ¯G/Im(α2) = ¯G/Ker(α3) ∼= Im(α3) = G
as required.
Any group extension must satisfy all the axioms of Proposition 3.1.2.
Example 3.1.1. Let us consider two groups, Z6 and S3. Both groups are extensions of Z3 by Z2.
However, Z6 is an extension of Z2 by Z3, but S3 is not such an extension as there is no normal
subgroup of index 3 in S3.
Example 3.1.2. Let N and G be any groups. Then the direct product N × G is an extension of N by G as well as G by N .
We end this section by remarking that, for any group extension ¯G of N by G, the group G does not necessarily exist as a subgroup of ¯G and if it does, it need not be unique. For instance, if we take ¯G = S6 and N = A6, then G can be any group of order 2.
3.2. Semi-Direct Products and Split Extensions
Definition 3.2.1. Let N and G be subgroups of a group ¯G such that 1. N E ¯G,
2. ¯G = N G, 3. N ∩ G = {1}.
Then ¯G is called a semi-direct product of N by G, denoted by N :G. In this case, the subgroups N and G are said to be complementary in ¯G. If N and G are both normal subgroups of ¯G, then
¯
G is a direct product of N and G.
For ¯G a semi-direct product of N by G, every element ¯g ∈ ¯G can be written uniquely as a product of an element of N by an element of G. To show uniqueness, assume that there exist n1, n2 ∈ N
and g1, g2∈ G such that ¯g1 = n1g1 and ¯g2 = n2g2. Then
¯ g1= ¯g2 ⇔ n1g1= n2g2 ⇔ g1g−12 = n −1 1 n2 ⇔ g1g−12 = 1G¯ ⇔ g1= g2 and n1= n2.
Definition 3.2.2. Let G and N be groups. Then the automorphism group of N denoted by Aut(N ), is the set of all automorphisms of N under the binary operation of composition.
For arbitrary finite groups N and G and for every group homomorphism ϑ defined as above, a unique semi-direct product, is determined. This semi-direct product is denoted by N :ϑG and
referred to as the semi-direct product of N by G, realized by ϑ.
Remark 3.2.3. Let N and G be arbitrary finite groups. Then a semi-direct product N :G exists if and only if there exists a homomorphism ϑ : G −→ Aut(N ) defined by ϑg(n) = gng−1 for all g ∈ G
and n ∈ N.
Remark 3.2.3 asserts that a semi-direct product is completely determined by a homomorphism ϑ : G −→ Aut(N ), that is to say, it is described by the way G acts on N . In cases where the action of G on N is well understood, we usually write N :G for N :ϑG to denote a semi direct product.
We note that if ¯G is an extension of N by G (see Definition 3.1.1), then β : ¯G −→ G is an epimorphism with Ker(β) = N. Define a lifting of g ∈ G to be an element ¯g ∈ ¯G by β(¯g) = g. Then by choosing a lift to each g ∈ G, we get the set {¯g : g ∈ G}, which is a transversal for N in G. Consequently, we also get a function φ : G −→ ¯G defined by φ(g) = ¯g called a transversal function. Suppose ¯g ∈ ¯G, the coset representative of N ¯g is φ(β(¯g)) = (φ ◦ β)(¯g). We further note that in general, the transversal function is not a homomorphism but always satisfies the relation (φ ◦ β)(¯g) = 1G¯ or similarly, (β ◦ φ)(g) = 1G¯.
Next, we discuss split and non-split extensions.
Definition 3.2.4. An extension {1} −→ N −→ ¯α G−→ G −→ {1} is split if there is a monomor-β phism φ : G −→ ¯G such that φ ◦ β = 1 (or equivalently if there is a transversal function φ which is a homomorphism). We say that an extension ¯G of N by G is non-split if it is not a split extension. We denote a non-split extension by ¯G = N·G.
Note that in the split extension case, ¯G contains G. If ¯G does not contain G (that is, contains only N ) then it is non-split.
The next result links split extensions to semi-direct products, it assures us that the two concepts are actually identical.
Proposition 3.2.5. (Scott [39]) The split extension {1} −→ N −→ ¯α G −→ G −→ {1} and theβ semi-direct product N :G are equivalent.
Proof. Let {1} −→ N −→ ¯α G−→ G −→ {1} be a split extension. Then there is a monomorphismβ φ : G −→ ¯G such that φ ◦ β = 1. Recall that β( ¯G) = G since β is an epimorphism and
Ker(β) = Im(α) = α(N ) = N . We now set
H = φ ◦ β( ¯G) = φ(G) = {φ(g) : g ∈ G}.
Since φ ◦ β = 1, we have that β(φ ◦ β(¯g)¯g−1) = β(¯g¯g−1) = 1, which implies that
φ ◦ β(¯g)¯g−1 ∈ Ker(β) = N and H = ¯G/N ∼= G.
Now since φ ◦ β(¯g) ∈ H, we obtain β(¯g) = β(φ ◦ β(¯g)) = 1 so that φ ◦ β(¯g) = φ(1) = 1 ∈ H. Thus N ∩ H = {1} and ¯G = N :H ∼= N :G. Therefore, every split extension is a semi-direct product. Conversely, every semi-direct product N :G is a split extension since the function φ : G −→ N :G, defined by φ(g) = (1N, g), is a transversal function which is a homomorphism.
Because of the equivalence established in Proposition 3.2.5, the study of semi-direct products is the same as the study of split extensions. Thus we shall denote a split extension of N by G, by N :G. In the following Lemma 3.2.6, we show that even for a non-split extension of N by G, if N is abelian, then G acts on N .
Lemma 3.2.6. (Rotman [36], Whitley [41]) If ¯G is an extension of N by G, with N abelian, then there is a homomorphism ϑ : G −→ Aut(N ) such that ϑg(n) = ng = ¯gn¯g−1 where n ∈ N, g ∈ G
and θ is independent of the choice of liftings {¯g : g ∈ G}.
Proof. Assume that x ∈ ¯G and δx denote conjugation by x. Since N is a normal subgroup of ¯G, we
have that (δx)N ∈ Aut(N ) and that ς : ¯G −→ Aut(N ) defined by ς(x) = (δx)N is a homomorphism.
Suppose x ∈ N, since N is abelian, we have that ς(x) = IN. Thus, there exists a homomorphism
ς∗ : ¯G/N −→ Aut(N ) given by ς∗(N x) = ς(x). But we know that G ∼= ¯G/N , hence for any transversal say {¯g : g ∈ G}, the function ψ : G −→ ¯G/N defined by ψ(g) = N ¯g is an isomorphism.
