A study of permutation groups and coherent configurations. John Herbert Batchelor. A Creative Component submitted to the graduate faculty

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1 A study of permutation groups and coherent configurations by John Herbert Batchelor A Creative Component submitted to the graduate faculty in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Major: Mathematics Program of Study Committee: Sung-Yell Song, Major Professor Irvin Hentzel Jue Yan Iowa State University Ames, Iowa 2009 Copyright c John Herbert Batchelor, All rights reserved.

2 ii TABLE OF CONTENTS ABSTRACT iii CHAPTER 1. HISTORICAL BACKGROUND Early Historical Background Resolvent Polynomials More Recent Developments Galois Groups Additional Facts CHAPTER 2. SHARPLY 2-TRANSITIVE GROUPS Notation and Basic Definitions Coset Spaces Characterization of Sharply 2-Transitive Permutation Groups CHAPTER 3. COHERENT CONFIGURATIONS Coherent Configurations and Basis Algebras Association Schemes Schur Rings Construction of Non-Symmetric Commutative Association Schemes Using Schur Rings Permutation Representations and Centralizer Algebras Character Theory BIBLIOGRAPHY ACKNOWLEDGEMENTS

3 iii ABSTRACT Permutation groups have fascinated mathematicians for hundreds of years; they are a central topic in abstract algebra. In this creative component, we will discuss permutation groups and their actions. We will begin by covering the historical background of the study of permutation groups, from Lagrange s results to more recent developments such as Galois groups. Next, we will discuss basic concepts and state definitions relevant to permutation group theory. This paper will also include some more advanced topics such as coherent configurations and association schemes. We will discuss Schur rings and permutation representations and conclude with a brief introduction to character theory.

4 1 CHAPTER 1. HISTORICAL BACKGROUND In this chapter, we will discuss the historical background relevant to the study of permutation groups, as well as some of the questions that have motivated the study of this topic. The purpose of this chapter is to provide a brief survey of the history of permutation group theory, with emphasis given to topics that will be covered later in this paper. Note that this chapter includes some references to terminology that will be defined in Chapter 2 and Chapter Early Historical Background Permutation groups were one of the first topics that mathematicians studied in group theory [5]. In 1770, Lagrange discussed permutations when he was studying algebraic solutions to polynomial equations. This problem involves determining the roots of a given polynomial in terms of an algebraic formula involving the polynomial s coefficients. Mathematicians wanted to derive formulas to construct roots using only addition, subtraction, multiplication, division, and extraction of k th roots (k N). This technique is called solution by radicals. Example The most familiar example of solution by radicals is the quadratic formula. Given a polynomial equation of the form ax 2 + bx + c = 0, the roots are given by x = b ± b 2 4ac. 2a Similar formulas exist for the solution of cubic and quartic (degree 4) equations. However, no such formula exists for quintic (degree 5) polynomials. Lagrange explained that these algorithms depended on finding resolvent polynomials, which are discussed in the next section. The roots of the resolvent polynomials are used to determine the roots of the original polynomials.

5 2 1.2 Resolvent Polynomials Consider a set A = {X 1, X 2,, X n } containing n variables. The symmetric group S n acts on A by permuting the subscripts of the variables. This action of S n on A can be extended to an action on the set of all polynomials in the variables X 1, X 2,, X n. Example Let z = ( 1 2 )( 3 4 ) S 4, and define Φ = X 1 X 3 X 2 X 4. Then Φ z = X 2 X 4 X 1 X 3 = Φ. The orbit of Φ under S 4 is given by O = {X 1 X 3 X 2 X 4, X 2 X 4 X 1 X 3, X 1 X 2 X 3 X 4, X 3 X 4 X 1 X 2, X 1 X 4 X 2 X 3, X 2 X 3 X 1 X 4 }. Lagrange called these six polynomials the values of Φ. By the orbit-stabilizer property (Theorem 2.1.1), the stabilizer of Φ in S 4 has order 4! 6 = 24 6 = 4. Definition Let Φ be a polynomial in X 1, X 2,, X n with k values Φ = Φ (1), Φ (2),, Φ (k). The resolvent h is a polynomial in X 1, X 2,, X n, Z defined as follows. h(z) = k ( Z Φ (i) ) = i=1 k ( ) h j X1, X 2,, X n Z j. j=0 Note that the values Φ (1), Φ (2),, Φ (k) form an orbit under S n, and so the polynomial h is invariant under arbitrary permutations of X 1, X 2,, X n. Therefore, each polynomial h j (X 1, X 2,, X n ) is symmetric in X 1, X 2,, X n, and we can express them as polynomials in the elementary symmetric functions of X 1, X 2,, X n. Example Let f(x) be a polynomial of degree n with roots r 1, r 2,, r n. Substitute r 1, r 2,, r n for X 1, X 2,, X n in the expression for h(z) to obtain the following polynomial in Z: k p(z) = h j (r 1, r 2,, r n )Z j. j=0 We can determine the coefficients of p(z) from the coefficients of f(x). By choosing Φ carefully, we may be able to solve the polynomial h(z) and determine the roots r 1, r 2,, r n from the roots Φ (1) (r 1, r 2,, r n ), Φ (2) (r 1, r 2,, r n ),, Φ (k) (r 1, r 2,, r n ) of h(z). Paolo Ruffini and Niels Abel used the method of resolvents to prove that solution by radicals cannot be used to solve the general equation of degree 5 or greater.

6 3 1.3 More Recent Developments theory. We close this chapter with a summary of some more recent results in permutation group Galois Groups Galois considered the question of whether a given polynomial could be solved using radicals. In 1830, he made a major discovery. For each polynomial f(x) with distinct roots r 1, r 2,, r n, there exists an associated permutation group on the set of roots. Mathematicians call this group the Galois group of f(x) [8]. The structure of the Galois group determines whether f(x) can be solved by radicals. Note that the following definition from [5] uses modern field theory terminology. Definition Consider a field K that contains the coefficients of f(x). Adjoin the roots to obtain a splitting field L. The field automorphisms of L that fix every element of K constitute a finite group G. This group acts on the set of roots. The Galois group of f(x) is the set of permutations of {r 1, r 2,, r n } induced by the elements of G. Next, we will discuss the relationship between Galois groups and Lagrange resolvents. Let Φ be a polynomial over K in n variables such that each of the roots r 1, r 2,, r n of f(x) can be expressed as a polynomial over K in t, where t = Φ(r 1, r 2,, r n ). That is, K(r 1, r 2,, r n ) = K(t). For each x S n, let t x = Φ(r 1, r 2,, r n ), where i = i x for all i {1, 2,, n}. Now define the resolvent g(z) as follows: g(z) = (Z t x ). x S n Note that g is a polynomial of degree n! over K. Factor g(z) over K to obtain an irreducible factor, say g 1 (Z), that has t as a root. Now let G denote the Galois group for f(x). Then g 1 (Z) has degree G, and the set of roots of g 1 (Z) is given by {t x : x G}.

7 Additional Facts Several other mathematicians have made major contributions to the development of permutation group theory. By the middle of the 1800s, mathematicians had developed a solid theory of permutation groups. In 1870, Camille Jordan published Traité des Substitutions et des Équations Algébriques. This book was based on papers written by Galois in It was the first text on the topic of permutation groups, and it was reprinted in William Burnside published important research on 2-transitive permutation groups. In 1897, he published his book, Theory of Groups of Finite Order. It was the first text on finite group theory written in English [4]. Burnside studied the problem of finding invariants of finite groups. He was particularly interested in the icosahedral group in four variables. He proved the following theorem on transitive permutation groups. Theorem Let G be a transitive permutation group on p symbols, where p is prime. Then G is either solvable or 2-transitive. Ferdinand Frobenius also contributed to the advancing study of permutation groups. In 1900, he published research on the use of permutation characters to compute the characters of the symmetric group S n. He served as Issai Schur s advisor, and he proved the following important result. Theorem Suppose that G is a transitive permutation group that acts on a set X containing n elements. Assume that no non-identity element of G fixes more than one element of X. Then there are exactly n 1 elements of G that have no fixed points in X. These n 1 elements, along with the identity, constitute a normal subgroup of G. Frobenius introduced the concept of Frobenius groups, a special kind of transitive group. A Frobenius group is defined as a transitive permutation group that is not regular (cf. Definition 2.1.9) but has the property that only the identity has two or more fixed points [5]. Frobenius, Zassenhaus, and J.G. Thompson proved important results about the structure of finite Frobenius groups. A 2-transitive Frobenius group is often called a sharply 2-transitive group (cf. Section 2.3). Examples of Frobenius groups include the dihedral groups of order 2q, where q

8 5 is odd [4]. In 1902, Frobenius proved that in a Frobenius group G of degree n, the identity together with the elements of degree n constitute a regular group which is a characteristic subgroup of G. Issai Schur specialized in representation theory of finite groups. He served as a mentor to Helmut Wielandt, who proved numerous results in permutation group theory [4]. In 1908, he published a new proof of Theorem Schur referred to this theorem as one of the most important new results in finite group theory. Schur expanded on some of Frobenius results concerning characters and representations of finite groups. His fundamental idea was to regard group elements as points, instead of using arbitrary objects or the integers 1, 2,, n [10]. Wielandt, Bercov, and Kochendörffer also proved important results concerning Schur rings. Coherent configurations, which are discussed in detail in Chapter 3, were introduced by Donald Higman in his 1967 paper, Intersection matrices for finite permutation groups. Higman defined the concept of structure numbers, denoted p h ij, which are described in Section 3.2. R.C. Bose and K.R. Nair discussed association schemes in their paper, Partially balanced incomplete block design. The study of association schemes grew from the fields of statistics and experimental design. Bose defined an association scheme as a coherent configuration (Ω, R) such that all relations in R are symmetric. Bose and D.M. Mesner described the basis algebra of an association scheme in 1959; this algebra is now called the Bose-Mesner algebra [3]. In 1973, Philippe Delsarte published his paper, An algebraic approach to the association schemes of coding theory, in which he proved several results concerning association schemes. Delsarte also developed the theory of duality for association schemes that admit a regular abelian automorphism group. Association schemes are discussed in detail in Section 3.2.

9 6 CHAPTER 2. SHARPLY 2-TRANSITIVE GROUPS In this chapter, we will discuss the basic concepts and definitions relevant to permutation groups. We shall conform to the standard notation used in Wielandt s book [10] and Cameron s book [3]. 2.1 Notation and Basic Definitions In this chapter we will study permutations on a finite set Ω. Let Ω be a finite set of arbitrary elements called points and denoted by lower case Greek letters such as α, β, γ, etc. Upper case Greek letters such as Γ, are used to denote subsets of Ω. For a subset Ω, let denote the number of points in, or the cardinality of. We will generally use positive integers to denote the points of Ω, as follows: Ω = {1, 2,, n}. A permutation on Ω is a one-to-one mapping of Ω onto Ω. The image of a point α Ω under a permutation g is denoted αg. Using this notation for the image, we express a permutation g as follows: g = 1 2 n = α 1g 2g ng αg The product of two permutations g and h on Ω, denoted gh, is defined as follows: α(gh) = (αg)h The identity permutation, denoted 1, leaves all points of Ω fixed. The permutation inverse to g, written g 1, is characterized by the following property: If g maps α to β, then g 1 maps β to α. Given a set Ω, the symmetric group S Ω is the set of all permutations on Ω. The group operation for S Ω is composition, denoted gh and defined by (α)gh = (αg)h, for all α Ω

10 7 and all g, h S Ω. When Ω = n, we often write S n to denote the symmetric group [7]. A permutation group G on a set Ω is a subgroup of S Ω. Next, we illustrate the cyclic form of a permutation with the following example = ( )( 2 )( 3 7 )( 6 ) = ( )( 3 7 ) The expression on the right is in reduced form, with cycles of length 1 omitted. We may write the identity permutation in cyclic form as follows. 1 = ( 1 ) Next, we note that every permutation can be decomposed uniquely into disjoint cycles, up to the ordering of the cycles [8]. For example, ( )( ) = ( 1 4 )( 2 5 ), under the convention (α)gh = (αg)h. A transposition t is a permutation that interchanges exactly two points and fixes all other points. Transpositions have the form t = ( α β ), where α, β Ω and α β. In [11], Zassenhaus observed that every n-cycle (cycle containing n points) may be expressed as a product of n 1 transpositions as follows: ( n ) = ( 1 2 )( 1 3 ) (1 n 2 )( 1 n 1 )( 1 n ). For instance, ( ) = ( 1 2 )( 1 3 )( 1 4 )( 1 5 ). An n-cycle is even if its decomposition into transpositions contains an even number of transpositions and odd if its decomposition contains an odd number of transpositions. Note that an n-cycle is even if and only if n is odd. A permutation g S n is even if its cycle decomposition contains an even number of cycles with an even number of points. Likewise, g is odd if its decomposition contains an odd number of cycles with an even number of points. Note that a permutation cannot be both even and odd [7]. The degree of a permutation group G 1 is the number of points moved by G. That is, the degree of G is the number of points not fixed by every g G. Cycles of even degree are odd permutations, and cycles of odd degree are even permutations. The order of a permutation is the least common multiple of the degrees of the cycles in its cyclic form. The alternating group on Ω, written A Ω (or A n if Ω = n), consists of all even permutations on Ω [6]. A permutation group G is cyclic if G is generated by a single permutation g. We denote this by G = g. The degree of a cyclic group G = g

11 8 is the degree of the permutation g. Let G be a group acting on a set Ω, and let α Ω. The set α G = {αg : g G} is called the orbit of G containing α. The stabilizer of α in G is the set G α = {g G : αg = α}. If G S Ω and Ω, the stabilizer of in G is the set G = {g G : αg = α, for all α }. If = {α}, then G = G α. A permutation group G is transitive if for all α, β Ω, there exists g G such that αg = β. So the action of G on Ω is transitive if there is only one orbit. The following result is called the orbit-stabilizer property. Theorem Let G be a finite permutation group on Ω. Then G = G α α G for all α Ω. Proof. Let α G = {β 1, β 2,, β r }. Then there exist g 1, g 2,, g r G such that αg i = β i for i = 1, 2,, r. Let P = {g 1, g 2,, g r }. Then P = α G. It is sufficient to show that for any g G, g can be expressed in exactly one way as the product of a permutation in P and a permutation in G α, so that G = P G α which will complete the proof. Given g G, αg = β k for some k {1, 2,, r}. This means that αg = αg k, so gg 1 k gg 1 k leaves α fixed. That is, G α. But (gg 1 k ) g k = g, so g is the product of a permutation in G α and a permutation in P. Now suppose that g can be written in two ways as such a product: g = hg k = fg l for h, f G α. Then αhg k = β k and αfg l = β l. This forces β k = β l, and so k = l. Therefore, g = h g k = f g k and h = f. The following orbit-counting lemma is often called Burnside s Lemma, but Burnside was not the first to prove it. It was known to earlier mathematicians including Frobenius. Let ψ(g) denote the number of points of Ω that are fixed by the element g G. That is, ψ(1) = Ω. Lemma Let G be a permutation group on a finite set Ω. Then the number of orbits of G on Ω is equal to the average number of fixed points of an element of G. That is, the number of orbits of G is Proof. Note that g G ψ(g) = α Ω 1 G ψ(g). g G G α = {(g, α) : αg = α}.

12 9 By the orbit-stabilizer property, 1 G g G ψ(g) = 1 G α = G α = 1 G G α G. α Ω α Ω α Ω The proof follows from the fact that the sum on the right-hand side indicates the number of orbits. Lemma Let G be a transitive permutation group on the finite set Ω with Ω > 1. Then there exists g G such that αg α, for all α Ω. (Such an element g is called fixed-point-free.) Proof. Note that this result follows from Burnside s Lemma, which states that the number of orbits of G on Ω equals the average number of fixed points of an element of G. The average number of fixed points is 1, and the identity fixes more than one point. So there exists some element g G that fixes no point. An alternate proof involving maximal subgroups is given below. We are given that G is a transitive permutation group on a finite set Ω, where Ω 2. We must show that there exists some g G such that αg α, for all α Ω. First, we claim that if g G, α Ω, and αg = β, then g 1 G α g G β. To see this, suppose that g g 1 G α g. Then g = g 1 hg, for some h G α. Note that αh = α (since h G α ). So βg = (αg)(g 1 hg) = α(gg 1 )hg = (αh)g = αg = β. That is, βg = β. Hence, g G β. Therefore, g 1 G α g G β. Our second claim is that gg α g 1 g G for α Ω is a proper subset of G. To show this, let M be a maximal subgroup containing G α in G. Consider the conjugation action of G on the set P(G) of all subgroups of G. We observe that the point stabilizer G M = M since G M = {g G : gmg 1 = M} = N G (M) and M is a maximal subgroup of G. This means that the size of the orbit containing M is G M by Theorem (the orbit-stabilizer property). Since each element of the orbit M G is a subgroup of G of

13 10 the form gmg 1 for some g G \ M, gmg 1 = M. As all subgroups contain the identity element of G, we have as desired. gg α g 1 g G g G gmg 1 ( M 1 ) G M < G, We now prove the conclusion by way of contradiction. Suppose that for all g G, there is an element α Ω such that αg = α. Let Ω = n (n 2, finite), and denote the elements of Ω by α 1, α 2,, α n. By transitivity, for all α i, α j Ω, there exists g ij G such that α i g ij = α j. By our assumption, for each g ij G, there is some element α k Ω such that α k g ij = α k. Hence, g ij G (αk ). So G = G (α1 ) G (α2 ) G (αn). By the first claim above, G (α1 ) G (α2 ) G (αn) = gg (α1 )g 1 ; g G so G = gg (α1 )g 1. g G This is a contradiction to the second claim, since G (α1 ) is a proper subgroup of G. This completes the proof. Definition Let k N be such that k < Ω. A permutation group G is k-transitive on Ω if G acts transitively on the set of k-tuples of distinct elements, where the componentwise action is defined by (α 1, α 2,, α k )g = (α 1 g, α 2 g,, α k g). In other words, G is k-transitive if for every two ordered k-tuples (α 1, α 2,, α k ) and (β 1, β 2,, β k ) consisting of distinct points of Ω, there exists g G such that α i g = β i for all i {1, 2,, k}. We say that G is sharply k-transitive if there exists exactly one such g G for every two such k-tuples. Note that if k 2, then G is k-transitive on Ω if and only if both of the following conditions hold: 1. G is transitive on Ω. 2. G α is (k 1)-transitive on Ω \ {α}. In some texts such as [10], this concept is called k-fold transitivity.

14 11 Definition [3] Let G be a transitive group on Ω. A block is a subset Ω with the property that for all g G, either g = or g =. The set Ω, the empty set, and the singleton sets {α} are blocks of each group G on Ω. They are called trivial blocks. Definition Let G be a transitive group on Ω. A congruence is an equivalence relation on Ω that is invariant with respect to G. The trivial congruences are the equality relation and the universal relation defined by α β, for all α, β Ω. All other congruences are called nontrivial congruences. Definition Let G be a transitive group. We say that G is imprimitive if there exists a nontrivial block. The block is called a set of imprimitivity. G is primitive if it has only trivial blocks. Using the terminology of Definition 2.1.6, a group G is primitive if it has only the trivial congruences and imprimitive if it has a nontrivial congruence. We state the following result first proved by Schur: Theorem Suppose that G is a permutation group on a set of n elements, where n N is not prime. Assume that G contains a cycle of order n. Then G is either 2-transitive or imprimitive. Definition Let G be a permutation group on Ω. G is semiregular if for every α Ω, G α = {1}. G is regular if it is both transitive and semiregular. In this case, we say that G acts regularly on Ω. 2.2 Coset Spaces In what follows, when G acts transitively on a finite set Ω, we will say that G is a transitive permutation group, or that Ω is a transitive G-space. We also say that two G-spaces Ω 1 and Ω 2 are isomorphic if there is a bijection θ : Ω 1 Ω 2 such that, for any α Ω 1 and g G, we have θ(αg) = θ(α) g. Let H be a subgroup of a finite group G. The (right) coset space H \ G is the set of right cosets H \ G = {Hx : x G}, with the action of G given by (Hx)g = Hxg, for all x, g G.

15 12 It is clear that H \ G is a transitive G-space. The classification of transitive G-spaces, up to isomorphism, is given by the following theorem from [3]. Theorem (a) Let Ω be a transitive G-space. Then Ω is isomorphic to the coset space H \ G, where H = G α for α Ω. (b) Two coset spaces H \ G and K \ G are isomorphic if and only if H and K are conjugate subgroups of G. Proof. (a) Let α, β Ω be given. It is routine to verify that the map f : Ω H \ G β {g G : αg = β} is an isomorphism, where the set {g G : αg = β} is a right coset of G α. (b) ( ) For each x G, the conjugate x 1 G α x of G α is the stabilizer of the point αx = β of Ω. Hence, G α \ G and G β \ G are isomorphic by (a). ( ) Let f be an isomorphism between H \ G and K \ G. If f(h) = Kx, then the stabilizer of H is the stabilizer of Kx, which is x 1 Kx (a conjugate of K). 2.3 Characterization of Sharply 2-Transitive Permutation Groups We will now consider some properties of finite sharply 2-transitive permutation groups (cf. Definition 2.1.4). The material in this section is loosely based on an exercise problem in [3]. Suppose that G is a finite sharply 2-transitive group on Ω, where Ω = n > 2. Then for every two 2-tuples (α 1, α 2 ) and (β 1, β 2 ) of elements of Ω with α 1 α 2 and β 1 β 2, there exists a unique element g G such that α 1 g = β 1 and α 2 g = β 2. That is, G acts regularly on the set of 2-tuples of distinct elements of Ω (cf. Definition 2.1.9). In the 1930s, Zassenhaus determined the finite sharply 2-transitive groups [12]. He showed that determining the finite sharply 2-transitive groups is equivalent to determining the finite near-fields. We will give a brief survey of this result. Definition A near-field is an algebraic structure consisting of a set F with two binary operations + (addition) and (multiplication) that satisfies the following axioms:

16 13 (i) (F, +) is an abelian group. (ii) For all x, y, z F, (x y) z = x (y z) (associative law for multiplication). (iii) For all x, y, z F, (x + y) z = x z + y z (right distributive law). (iv) There exists an element 1 F such that for all x F, 1 x = x 1 = x (multiplicative identity). (v) For each nonzero element x F, there exists x 1 F such that x x 1 = 1 = x 1 x (multiplicative inverses). Note that every field is a near-field. Zassenhaus classified the finite near-fields that are not fields [3]. Most such near-fields are Dickson near-fields. They are obtained from fields by using field automorphisms to twist the multiplication. There are also seven exceptional near-fields with orders 5 2, 7 2, 11 2, 11 2, 23 2, 29 2, and 59 2, respectively. Definition Let F be a near-field with F = n, and let F = F \ {0}. Define A = {τ a,b : (a, b) F F, and τ a,b (x) = xa + b for all x F }. We call A the group of affine transformations of the near-field F. Note that A = F F = (n 1)n. Proposition Given a near-field F, the group A of affine transformations is a sharply 2-transitive group on F. Proof. It is routine to verify that A is a transitive permutation group on F. We claim that A acts sharply 2-transitively on F. To see this, let (x 1, y 1 ), (x 2, y 2 ) F F with x 1 y 1 and x 2 y 2. Then the system of two equations τ a,b (x 1 ) = x 1 a + b = x 2 and τ a,b (y 1 ) = y 1 a + b = y 2 has a unique solution a = (x 1 y 1 ) 1 (x 2 y 2 ) and b = x 2 x 1 a = x 2 x 1 (x 1 y 1 ) 1 (x 2 y 2 ). That is, there exists a unique transformation τ a,b = g A such that x 1 g = x 2 and y 1 g = y 2. Therefore, A is a sharply 2-transitive group on F.

17 14 Example Let F = Z 5 = {0, 1, 2, 3, 4}. So n = F = 5. Then A = {τ 1,0, τ 1,1, τ 1,2, τ 1,3, τ 1,4, τ 2,0, τ 2,1, τ 2,2, τ 2,3, τ 2,4, τ 3,0, τ 3,1, τ 3,2, τ 3,3, τ 3,4, τ 4,0, τ 4,1, τ 4,2, τ 4,3, τ 4,4 }. Note that τ 1,0 = 1 (identity) since for all x F, τ 1,0 (x) = 1 x + 0 = x. The transformations τ 1,1, τ 1,2, τ 1,3, τ 1,4 are the translates x x + 1, x x + 2, x x + 3, x x + 4, respectively, and have no fixed points. The point stabilizers are A 0 = {τ 1,0, τ 2,0, τ 3,0, τ 4,0 }, A 1 = {τ 1,0, τ 2,4, τ 3,3, τ 4,2 }, A 2 = {τ 1,0, τ 2,3, τ 3,1, τ 4,4 }, A 3 = {τ 1,0, τ 2,2, τ 3,4, τ 4,1 }, and A 4 = {τ 1,0, τ 2,1, τ 3,2, τ 4,3 }. We note that each of the 15 elements in A i \{τ 1,0 } for i {0, 1, 2, 3, 4} fixes exactly one point, while 4 translates τ 1,1, τ 1,2, τ 1,3, τ 1,4 fix no points and the unique element 1 = τ 1,0 fixes all points of F = Z 5. The set H consisting of the identity and the elements that fix no points is a subgroup of A. The group table for H = {τ 1,0, τ 1,1, τ 1,2, τ 1,3, τ 1,4 } is given below. τ 1,0 τ 1,1 τ 1,2 τ 1,3 τ 1,4 τ 1,0 τ 1,0 τ 1,1 τ 1,2 τ 1,3 τ 1,4 τ 1,1 τ 1,1 τ 1,2 τ 1,3 τ 1,4 τ 1,0 τ 1,2 τ 1,2 τ 1,3 τ 1,4 τ 1,0 τ 1,1 τ 1,3 τ 1,3 τ 1,4 τ 1,0 τ 1,1 τ 1,2 τ 1,4 τ 1,4 τ 1,0 τ 1,1 τ 1,2 τ 1,3 We observe that τ 1,α τ 1,β = τ 1,(α+β)(mod 5), for all α, β Z 5. Therefore, the group (H, ) is isomorphic to an additive group of order 5. The point stabilizer A 1 = {τ 1,0, τ 2,4, τ 3,3, τ 4,2 } for the element 1 F is also a subgroup of A. The group table for A 1 is given below.

18 15 τ 1,0 τ 2,4 τ 3,3 τ 4,2 τ 1,0 τ 1,0 τ 2,4 τ 3,3 τ 4,2 τ 2,4 τ 2,4 τ 4,2 τ 1,0 τ 3,3 τ 3,3 τ 3,3 τ 1,0 τ 4,2 τ 2,4 τ 4,2 τ 4,2 τ 3,3 τ 2,4 τ 1,0 Proposition Every finite sharply 2-transitive group G on Ω is realized as the affine transformation group A of a near-field F such that G = A and Ω = F. Proof. First, we will show that every sharply 2-transitive group G has an abelian regular normal subgroup. We observe that G contains no element except the identity that fixes more than one point of Ω = {ø 1, ø 2,, ø n }. To see this, suppose that an element g G fixes two distinct points ø i, ø j Ω. Then ø i g = ø i and ø j g = ø j. Therefore, (ø i, ø j )g = (ø i, ø j ). By sharp 2-transitivity, it follows that g = 1, the identity of G. Next, we will prove that G contains n(n 2) elements that fix exactly one point. Given two 2-tuples (ø 1, ø 2 ) and (ø 1, ø i ) with i {3, 4,, n}, there exists a unique element g i G such that (ø 1, ø 2 )g i = (ø 1, ø i ). The group elements g i, g j must be different for i, j {3, 4,, n} whenever i j. So the stabilizer G ø1 of ø 1 contains exactly n 2 non-identity elements. Clearly, this number is the same for all ø i Ω. We have that G øi G øj = {1} whenever i j. Since the sets G ø1 \ {1}, G ø2 \ {1},, G øn \ {1} are disjoint, there exist exactly n(n 2) elements of G that fix exactly one point. By the above remark, the length of each orbit ø G i is n. That is, ø G i = n, for all i {1, 2,, n}. By the orbit-stabilizer formula, G = G øi ø G i = (n 1) n. That is, G = n(n 1). Only the identity fixes more than one point. We have that n(n 2) elements fix exactly one point, and one element (the identity) fixes all points. Therefore, Number of elements with no fixed point = n(n 1) n(n 2) 1 = n 1. So there exist n 1 elements of G that fix no point. Now let A denote the set consisting of the n 1 fixed-point-free elements and the identity of G. Note that A = (n 1) + 1 = n = Ω. We will prove that the set A is a regular normal subgroup of G. Note that for all x, y A, the element xy is fixed-point-free. Therefore, A is a subgroup of G. To see that A is a regular

19 16 subgroup (i.e., transitive and semiregular), first note that A = Ω. This implies that A is semiregular if and only if it is regular. We claim that for any i, j {1, 2,, n}, there exists at most one g A such that ø i g = ø j. To see this, note that if ø i g = ø j and ø i h = ø j, then ø i gh 1 = ø i. That is, A contains an element gh 1 that fixes ø i. Since all elements of A except the identity are fixed-point-free, we obtain that gh 1 = 1. Therefore, g = h. Therefore, A is a regular subgroup of G. To see that A is a normal subgroup of G, we must show that x 1 Ax = A, for all x G. It is sufficient to show that x 1 Ax A, for all x G. Note that for any element g A, the cycle structure of a conjugate of g is the same as that of g. So x 1 gx is fixed-point-free in G, for all g A \ {1} and all x G. Hence, x 1 Ax A, for all x G. It follows that x 1 Ax = A, for all x G. Therefore, A G. We obtain that A is a regular normal subgroup of G. Let x G be a fixed-point-free element. Then αx α, for all α Ω. Consider the centralizer C G (x) = {y G : x 1 yx = y} = {y G : xy = yx}. The fixedpoint-free element x G commutes with all fixed-point-free elements of G. So C G (x) = A, for any fixed-point-free element x G. As noted above, if x, y are distinct fixed-point-free elements of G, then xy is fixed-point-free. Let x, y G be fixed-point-free elements. Then (x 1 yx)(α) = y(α), for all α Ω. So xy = yx, and x 1 yx is fixed-point-free. Hence, the set C G (x) = {y G : xy = yx} contains all fixed-point-free elements of G. Since G contains n 1 fixed-point-free elements, it follows that x has at least n 1 conjugates. Therefore, every fixed-point-free element has at least n 1 conjugates. We obtain that the fixed-point-free elements and the identity form an abelian regular normal subgroup of G. Hence, every sharply 2-transitive group G has an abelian regular normal subgroup A. Next, we will build a near-field F with additive group A and multiplicative group G α, by identifying Ω with A (so that α corresponds to 0) and identifying Ω\{α} with the regular group G α. This subgroup A is a group of order n, with composition of permutations as its operation. We have that for all ø Ω, the point stabilizer G ø is of order n 1. We now build a near-field F as follows. Begin with an additive group F = (F, +) that is isomorphic to A. Set F = Ω, and identify 0 F with ø Ω. Next, define a multiplication on F = F \ {0} such that (F, ) is isomorphic to G ø. We observe that (F, ) = (F \ {0}, ) = G ø = {g G : øg = ø}.

20 17 Suppose that (a, b) = (g, h) with a, b, g, h F \ {0}. Then a = g and b = h. So a b = g h. Therefore, the multiplication is well-defined. Finally, we must show that the affine transformation group A of F is isomorphic to G. The elements of G α \ {1} fix only α. This suggests that G behaves like an affine transformation group of a near-field F, where F may be defined as F = A with its addition operation as the group operation in G and the multiplication operation on F \ {0} as the group operation on A \ {0}. To make this precise, note that (F, +) = A, where 0 denotes the identity of F and 1 denotes the identity of A. For all α Ω, G α contains n 1 elements including the identity. So (F \ {0}, ) = G α, where α Ω corresponds to 0 F. We obtain that (F, +, ) is a near-field. Now construct the affine transformation group A based on F : A = {(a, b) F F : x ax + b}. The action of A on F corresponds to the action of G on Ω. Therefore, A = G. We conclude that every finite sharply 2-transitive group G on Ω is realized as the affine transformation group A of a near-field F with the property that G = A and Ω = F.

21 18 CHAPTER 3. COHERENT CONFIGURATIONS In this chapter, we will discuss some additional topics relevant to the study of permutation groups. 3.1 Coherent Configurations and Basis Algebras Next, we will discuss the concept of a coherent configuration and the associated basis algebra. Let G be a permutation group acting on a set Ω. Consider the mapping from G (Ω Ω) to Ω Ω given by ( g, (α, β) ) (αg, βg), for all g G and all α, β Ω. The orbits of this action are called 2-orbits (or orbitals) of G on Ω. We have that (α, α) G = {(αg, αg) : g G} {(α, α) : α Ω}. Depending on G, the set (α, α) G may be a proper subset of {(α, α) : α Ω}. For i, j {1, 2,, n}, define (α i, α j ) G = {(α i g, α j g) : g G}. Case 1 (homogeneous case): If G acts on Ω transitively, then (α i, α i ) G = {(α, α) : α G} for all i {1, 2,, n}. Case 2 (non-homogeneous case): If G acts non-transitively on Ω, then the diagonal relation is partitioned into multiple 2-orbits. Definition Let R 1, R 2,, R t be the 2-orbits of G on Ω, with the following properties: 1. Ω Ω = R 1 R 2 R t (disjoint union). That is, {R 1, R 2,, R t } is a partition of Ω Ω. 2. For all i {1, 2,, t}, there exists i {1, 2,, t} such that R T i = {(β, α) : (α, β) R i } = R i.

22 19 3. For all i, j {1, 2,, t} and all α, β R h, the cardinality p h ij = {γ Ω : (α, γ) R i and (γ, β) R j } depends only on h, i, j and not on the choice of α, β. Let R = {R 1, R 2,, R t }. The pair (Ω, R) is called a coherent configuration. More precisely, (Ω, R) is the coherent configuration associated with G. It is called a homogeneous coherent configuration if the action of G on Ω is transitive (see Case 1 above). Definition For each orbital R i with 1 i t, the corresponding adjacency matrix A i is defined by 1, if (x, y) R i (A i ) xy = 0, if (x, y) / R i The rows and columns are indexed by the elements of Ω. Definition Let A = {A 1, A 2,, A t } be the set of adjacency matrices for the given coherent configuration (Ω, R). Let V (A) denote the span of A over the field of complex numbers. V (A) is called the basis algebra of the coherent configuration. 3.2 Association Schemes In what follows, we will consider homogeneous coherent configurations, which are also known as association schemes. We will use additive notation for abelian groups and multiplicative notation for nonabelian groups. Let G be a finite group, and define G as the set of mappings g : G G such that x xg = g(x), for all x G. Note that G = G. The permutation group G is called the right regular representation of G by its right translation. The action of G on G is transitive. Consider the mapping from G (G G) to G G defined by ( g, (x, y) ) (xg, yg). Let R be an orbital of the permutation group G on G. If (x, y) R and yx 1 = g, then R = {(x, y)h : h G} = {(xh, yh) : h G} = {(a, b) : (a, b) G G, ba 1 = g}.

23 20 Note that if (x, y)h = (xh, yh) = (a, b), then ba 1 = (yh)(h 1 x 1 ) = yx 1 = g. Conversely, suppose that (a, b) G G with ba 1 = g. So b = ga. Let h = x 1 a. Since a, x G, it follows that h G. Then (xh, yh) = (xx 1 a, yx 1 a) = (a, gxx 1 a) = (a, ga) = (a, b). Therefore, there exists h G such that (xh, yh) = (a, b). The orbital containing (x, y) where yx 1 = g contains all pairs (a, b) with ba 1 = g. This means that there is a bijection between the set of orbitals and G. Denote the orbital R = {(a, b) : ba 1 = g} by R g. It is shown that X(G) = ( G, {R g } g G ) is an association scheme. For a given group G, X(G) = ( G, {Rg } g G ) has the following properties, as discussed in [2]. 1. The diagonal relation is given by R 1 = {(a, a) : a G} G G, where 1 is the identity of G. Also, R g = G G g G and R g R h = whenever g h. So the orbitals R g form a partition of G G. 2. For all g G, R T g = {(y, x) : (x, y) R g } = R g 1. Note that if yx 1 = g, then (yx 1 ) 1 = xy 1 = g 1, and so (y, x) R g 1. So the symmetric conjugate of every relation in {R g } g G also belongs to {R g } g G. 3. For all h, i, j G and all (x, y) R h, the structure number p h ij = {z G : (x, z) R i and (z, y) R j } is constant (i.e., does not depend on the choice of (x, y) from R h ). Lemma Let G = {1 = g 0, g 1, g 2,, g d } be a group, and let X(G) = ( ) G, {R g } g G be the association scheme described above. Then for all i, j {0, 1,, d}, (a) p h ij = 0 or ph ij = 1. (b) p h ij = ph ji if and only if G is abelian.

24 21 Proof. (a) Note that for given x, y, g i, and g j, there exists a unique z such that z = g i x and there exists a unique z such that z = gj 1 y. Since p h ij = {z G : zx 1 = g i and yz 1 = g j } = {z G : z = g i x and z = gj 1 y}, we obtain that the cardinality p h ij is 1 if and only if z = z. Otherwise, p h ij = 0. (b) ( ) Assume that G is abelian. Let g i, g j, g h G, and let x, y be such that yx 1 = g h. If p h ij = 1, then there exists z G such that z = g ix = g 1 j y. This implies that g i x = gj 1 y or equivalently y = g j g i x. Since G is abelian, g i g j = g j g i. So y = g i g j x. Hence, g 1 i y = g 1 i (g i g j x) = g j x. If we set w = gi 1 y = g j x, then wx 1 = g j and yw 1 = g i. Therefore, p h ji = 1. A similar argument shows that ph ji = 1 implies ph ij = 1. ( ) Suppose that p h ij = ph ji for all i, j {0, 1,, d}. By way of contradiction, assume that G is not abelian. Choose g i, g j G such that g j g i g i g j. Then gi 1 g j g i g j. There exists g h {g 0, g 1,, g d } such that p h ij = 1. So there exists z = g ix = gj 1 y for some x, y G such that yx 1 = g h. But g 1 i that wx 1 = g j and yw 1 = g i. Hence, p h ij abelian. y = g 1 g j g i x g j x, and so there is no element w G such i = 0. This is a contradiction. Therefore, G is Here is another example of an association scheme coming from a finite group. Example Given G, the product G G acts transitively on G by the conjugation action x x(g 1, g 2 ) = g 1 1 xg 2, for x G and (g 1, g 2 ) G G. This action is transitive because for all x, y G, there exists (g 1, g 2 ) = (x, y) such that x(x, y) = x 1 xy = y. Let R 0, R 1,, R d be orbitals defined by R i = {(x, y) : yx 1 C i } for a conjugacy class C i of G. Given (g 1, g 2 ) G G and (x, y) G G, (x, y) ( x(g 1, g 2 ), y(g 1, g 2 ) ) = (g 1 1 xg 2, g 1 1 yg 2). So (g 1 1 yg 2)(g 1 1 xg 2) 1 = (g 1 1 yg 2)(g 1 2 x 1 g 1 ) = g 1 1 yx 1 g 1 C i. Letting z = g 1 1 xg 2 and w = g 1 1 yg 2, we obtain that (z, w) R i wz 1 C i.

25 22 Hence, we conclude that the association scheme obtained above by the action x g 1 1 xg 2 for all x G and all (g 1, g 2 ) G G is isomorphic to the scheme defined by letting R i = {(x, y) : yx 1 C i }, where {e} = C 0, C 1, C 2,, C d are conjugacy classes of G. We are now prepared to discuss some properties of association schemes. Let X = ( X, {R i } 0 i d ) be an association scheme of class d. We state the following axioms of an association scheme: 1. A 0 + A A d = J, the all-ones matrix. 2. For all i {0, 1,, d}, there exists i {0, 1,, d} such that A T i = A i. 3. For all i, j {0, 1,, d}, The span A i A j = d p h ija h = p 0 ija 0 + p 1 ija p d ija d. h=0 { d } A 0, A 1,, A d = a i A i : a i C is a (d + 1)-dimensional subspace of M X (C). This subspace is closed under multiplication and forms a ring. Definition The matrix algebra A = I = A 0, A 1, A 2,, A d is called the Bose-Mesner algebra of X. i=0 3.3 Schur Rings In this section, we will discuss a concept attributed to Issai Schur ( ), whose work in group theory and representation theory is chronicled in [4]. Some of the material in this section is adapted from{ Chapter 4 of [10]. } To see another presentation of X(G), consider the group algebra C(G) = c g g : c g C formed by the formal series c g g, c g C, with g G g G { } formal basis element g G. The subring Z(G) = a g g : a g Z is called a group ring g G over G. Later we will define the concept of a Schur ring, which is a particular subring of Z(G). Definition Let G be a group, and let e denote the identity element of G. In the group ring Z(G), the element T = g for any subset T G is called a simple quantity. g T

26 23 Definition A subring S of the group ring Z(G) is called a Schur ring (S-ring) over G if S is a Z-module having a basis {T 0, T 1,, T d } of simple quantities with the following properties: (S1) {e} = T 0, T 1,, T d form a partition of G, and (S2) For each i {0, 1, 2,, d}, T 1 i = {g 1 : g T i } = T i for some i {0, 1, 2,, d}. A set of simple quantities T 0, T 1,, T d that gives rise to the Schur ring S = T 0, T 1,, T d is called the standard basis for S. In this case, the integers p h ij such that d T i T j = p h ijt h for i, j {0, 1,, d} h=0 are called the structure numbers of S. Lemma implies that for all i, j {0, 1,, d}, d d T i T j = p h ijt h = p h jit h = T j T i. h=0 h=0 That is, T i T j = T j T i for all i, j {0, 1,, d}. Here (g, h) R i if and only if hg 1 T i. This implies that the scheme X(G) = ( ) G, {R i } 0 i d is commutative if and only if the group G is abelian. Note that in the examples below, (g, h) R i h g T i (since the groups are additive). Example Consider the cyclic group Z 4 = {0, 1, 2, 3} and the group ring Z(Z 4 ) = {a a a a 3 3 : a i Z} = 0, 1, 2, 3. Define the Z-module S 1 = 0, 1, 3, 2. So T 0 = {0}, T 1 = {1, 3}, and T 2 = {2}. The following calculation in Z(Z 4 ) (using simple quantity notation) shows that S 1 is an S-ring over Z 4. 1, 3 1, 3 = , 3 2 = 1, = 0. The S-ring S 1 is called the symmetrization of Z(Z 4 ). Example Let G = Z 6 = {0, 1, 2, 3, 4, 5}. Consider the following three Z-modules: S 2,1 = 0, 1, 3, 2, 4, 5 S 2,2 = 0, 2, 3, 4, 1, 5 S 2 = 0, 1, 4, 2, 5, 3.

27 24 The first two Z-modules are not S-rings. Note that S 2,1 does not satisfy the condition (S2), while S 2,2 is not a subring of Z(Z 6 ). To see this, note that 1, 5 1, 5 = / S 2,2. The third Z-module S 2 is an S-ring, as shown by the following calculation. Note that in S 2, we have T 0 = {0}, T 1 = {1, 4}, T 2 = {2, 5}, and T 3 = {3}. 1, 4 1, 4 = 2 2, 5, 1, 4 3 = 1, 4, 1, 4 2, 5 = , 3 3 = 0, 3 2, 5 = 2, 5, 2, 5 2, 5 = 2 1, 4. Definition A commutative association scheme is a coherent configuration such that all basis matrices commute. A symmetric association scheme is a coherent configuration with the property that all of the basis matrices are symmetric. Lemma Every symmetric association scheme is commutative. Proof. Let X be a symmetric association scheme. We consider the adjacency matrices A 0, A 1,, A d in the Bose-Mesner algebra of X. Since X is symmetric, we have that A T i = A i for all i {0, 1,, d}. Let i, j {0, 1,, d} be arbitrary. Then A i A j = A T i A T j (since X is symmetric) = (A j A i ) T (by properties of the transpose operator) = A j A i (since the product of two symmetric matrices is symmetric). That is, A i A j = A j A i. So A i A j = A j A i, for all i, j {0, 1,, d}. Therefore, X is a commutative association scheme. 3.4 Construction of Non-Symmetric Commutative Association Schemes Using Schur Rings In this section, we will show how Schur rings can be used to produce examples of commutative association schemes. The first example we will consider is Z(Z 4 ). In particular, Z(Z 4 ) represents a 3-class nonsymmetric commutative association scheme [9]. We will present four examples of non-symmetric commutative association schemes using three different notions: a relation graph, a scheme relation matrix, and a corresponding S-ring. By definition,

28 25 the scheme relation matrix A is given by A xy = i if and only if (x, y) R i, for the scheme X = ( ) X, {R i } 0 i d. That is, A = A1 +2A 2 + +da d, where A i denotes the adjacency matrix of (X, R i ). The directed vertex-edge graphs corresponding to the scheme relation matrices are also shown in the following examples. Example (Example Revisited) A ( X(Z 4 ) ) = = , where A 1 = , A 2 = , and A 3 = So A ( X(Z 4 ) ) = A 1 + 2A 2 + 3A 3, and Z(Z 4 ) = 0, 1, 2, 3. The graph represented by the adjacency matrix A 1 is given above. Note that 1 1 = 2 and 3 3 = 2.

29 26 Example (Example Revisited) A(X 2 ) = = A 1 + 2A 2 + 3A 3 ; S 2 = 0, 1, 4, 2, 5, The graph represented by the adjacency matrix A 1 is given above. Note that 1, 4 1, 4 = 2 2, 5 and 2, 5 2, 5 = 2 1, 4. Example Let F = , I 7 = Y = , J 2 = ,

30 27 Consider the matrix A(X 3 ) = I 7 Y + F J 2. This matrix is given by A(X 3 ) = = A 1 + 2A 2 + 3A

31 28 The graph represented by the adjacency matrix A 1 is given above. The Z-module S 3 = 0, 1, 2, 4, 8, 9, 11, 3, 5, 6, 10, 12, 13, 7 is an S-ring. To see this, note that 1, 2, 4, 8, 9, 11 1, 2, 4, 8, 9, 11 = 2 1, 2, 4, 8, 9, , 5, 6, 10, 12, 13, 3, 5, 6, 10, 12, 13 3, 5, 6, 10, 12, 13 = 2 3, 5, 6, 10, 12, , 2, 4, 8, 9, 11. Example Consider the quaternion group Q 8 using multiplicative notation: Q 8 = {1, i, j, k, i, j, k, 1}

32 29 The scheme relation matrix A(X 4 ) is given by 1 1 i i j j k k i A(X 4 ) = i j j k k = A 1 + 2A 2 + 3A 3. The graph represented by the adjacency matrix A 1 is given above. The Z-module S 4 = 1, i, j, k, i, j, k, 1 is an S ring. To see this, note that i, j, k i, j, k = i, j, k + i, j, k + 3 1, i, j, k i, j, k = i, j, k + i, j, k

33 Permutation Representations and Centralizer Algebras The relationship between groups and their representations is covered in many books including [1]. We will now recall the important concepts of permutation representations and centralizer algebras. Let G be a permutation group on Ω = {ω 1, ω 2,, ω n }. Then each element g G can be represented by a permutation matrix P (g), an n n matrix of zeros and ones, having (i, j)-entry 1, if ω i g = ω j P (g) i,j = 0, otherwise. The map P : g P (g) is a group homomorphism from G to the general linear group GL(n, C), the group of n n nonsingular matrices over the field of complex numbers. The matrix P (g) is called the permutation matrix corresponding to g. The permutation g = ω 1 ω 2 ω n ω 1 g ω 2 g ω n g S Ω induces the linear transformation [ω 1, ω 2,, ω n ]P (g) T = [ω 1 g, ω 2 g,, ω n g]. The homomorphic image P (G) consisting of all matrices P (g) for g G is called the permutation representation of G. Definition [2] The centralizer algebra of G is the set of n n matrices that commute with all of the permutation matrices P (g) for g G. 3.6 Character Theory Suppose that G is a finite group. We define a matrix representation of G of degree n to be a homomorphism M : G GL(n, C). So M(gh) = M(g)M(h), for all g, h G. Definition Let M, N be two matrix representations. We say that M and N are equivalent if N(g) = X 1 M(g)X for all g G, where X denotes an invertible matrix. In other words, M and N are equivalent if they are related by a change of basis.

34 31 A representation M is said to be reducible if it is equivalent to a representation of the form M 1(g) 0. M 2 (g) We say that M is irreducible if it is not equivalent to such a representation. The matrix representation M is decomposable if it is equivalent to a representation M 1(g) 0. 0 M 2 (g) If M is not equivalent to such a representation, we say that it is indecomposable. Definition The character of a representation M is the function χ : G C defined by χ(g) = Trace M(g), for all g G. The following theorem describes the relationship between matrix representations and characters. Theorem (a) Every character is constant on the conjugacy classes of G. (b) Two matrix representations are equivalent if and only if they have the same character. A character χ is said to be irreducible if it is the character of an irreducible representation [2]. The degree of a character χ is equal to the degree of the corresponding representation. We can express any character as a sum of irreducible characters. The degree of any character χ is equal to χ(1). The principal character of a group is the character of degree 1 that maps all group elements to the number 1 C. For each element g G, the permutation character π is given by π(g) = ψ(g) (number of points fixed by g) = number of ones on the diagonal of G = trace of permutation matrix P (g). Let Irr(G) denote the set of irreducible characters of G. We can decompose the permutation character π into irreducible characters as follows: π = χ Irr(G) m χ χ.