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Construction of Character Tables of Symmetric Group of Degree 3 and 4 (S 3, S 4,)

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Construction of Character Tables of Symmetric Group of Degree 3 and 4 (S 3, S 4,) Scholarly Journal of Mathematics and Computer Science Vol. 4(1), pp. 8-12, July 2015 Available online at http:// www.scholarly-journals.com/SJMCS ISSN 2276-8947 © 2015 Scholarly-Journals Full Length Research Paper Construction of character tables of symmetric group of degree 3 and 4 (S 3, S 4,) Isa Sani and Tijjani Bukar 1Federal University Gashua. Nigeria 2Department of Mathematics Yobe State University Damaturu Nigeria Accepted 13 July, 2015 Since most problems in representation of finite groups are best solved by character tables, we have constructed character tables of symmetric group of degree 3 and 4 (S 3 and S 4). Lifting process and orthogonality relations were applied in deriving the solutions. The results revealed that the symmetric group of finite representation can be completely reducible using character tables. Key words: Symmetric group, orthogonality, finite representation. INTRODUCTION In mathematics, especially the field of algebra which simple group cannot have a generalized quartenim group deals with group theory, the character of a representation However, one may see that one dwelt so much on the plays a very important role. It is a function on the group as its sylow – 2 – subgroups. Background of characters, which associates to each group element the trace of the this is because it will enable us acquire enough corresponding matrix. Characters were introduced by information that will guide us in the construction of George Frobenius (1849-1917). He initially developed various character tables. Thus, it is possible the one may representation of finite groups entirely based on study character and representatives of group with one characters. This is possible because a complex knowing much about character tables, but it is not representation of a group is determined up to possible to construct character table without enough (Isomorphism) by its characters. Group representation knowledge on character theory. Now for proper over a field of positive characteristics which is called understanding of this paper we need to explain some (Modular Representation) is very delicate, but Richard terminologies used. Moreover, there are some basic Brauer developed a very useful theory of characters theorem corollaries and lemma that we use to back our which helps a lot in this case as well. Hence, many deep points. theories on the structure of finite groups use the character of modular representations. Serre (1979), Definition 1 (Baker (2011)) stated that, characters are easy to tabulate and compute with, using character tables. A representation of a group G over a field F is a They expressed deep properties of irreducible homomorphism φ from G to GL (nF) [an nxn matrix with representations. Therefore they are essential tools in the entries in F) for some n, and the degree of φ is the integer classification of finite groups. Close to half of the proof of n. Therefore, if φ is a function from G → Gl (nF), then φ is Walter Felt Thompson’s theorem involve indicate a representation if and only if φ (gh) = φ (g) φ(h) Vg, h ∈G. calculations with character values. The Burnside theorems, Michio Suzuki, all use a lot of character theory Definition 2 (Momoh, 2010) in providing their theorems. For example, a theorem of Richard Brauer and Michio Suzuki stating that a finite Let G be a finite group and V a rector space over a field F. A homomorphism φ of G into the group GL (VF) of non singular transformation of V is called a representation of G. Therefore φ is a representation of G if for all gh ∈G *Corresponding author’s e-mail: [email protected]. Sani and Bukar 9 then φ(gh) = φ(g) (h), and the image of φ, im φ = GL (VF). over F. Furthermore, the G-module V over F is said to be completely reducible if V = U 1 ⊕ U2 ⊕… ⊕ Ui. Where U i ⇒ l=1, 2…1 are irreducible G-module over F. Equivalence Representation Definition 7 (Reducible representation Matrix Let φ: G → GL (mf) and θ:G φ GL (nF) be 2 Version) representation of G over F. we say that φ is equivalent to θ if n = m and there exists invertible nxn matrix T such The matrix representation A(x) is irreducible over F if ∃ a that for all g belongs to G, θ(g) = T -1 φ (g) T. (FG – non-singular matrix T over F such that Module): B(X) = T 1 A (X) T = C(x) 0 for all x ∈G Definition 3 E(x) D(x) Let V be a rector space over a field F and let G be a Where if A(X) ∈ GL (m), C(x) an r x r, D(X) an S x S group. Then V is an FG-module if multiplication Vg (for all Then E (x) is an S x r and m = str v∈V, g ∈G) is defined satisfying the following conditions. For all uv ∈ V, λ ∈ F and V gh ∈G, then We assume that the reader is familiar with Group theory and its properties, Group homomorphism, normal i. V ig∈G subgroups, conjugacy, cosets of a group and their likes, ii. V i (gh) = V ig) h because this will be relevant in our field of study of iii. V i I = V I character theory and construction of the character tables. iv. ( λ1V1 + λ2V2 +…. + λnVn) g = λ1(V 1g) + λ2 (V 2g) + …+ λn (Vng) RESULTS Definition 4 The irreducible characters of finite groups G are class (1). An FG-module V is said to be irreducible or simple if functions and the number of them is equal to the number it is non-zero and it has no FG-sub module apart from {0} of the conjugacy classes of G. it is convenient to record and V. all these values of all irreducible characters of G in a (2). An FG-module which is a direct sum of simple sub square matrix. The matrix is called the character table of modules is said to be completely reducible or semi- G. The entries in a character table are related to each simple other in a subtle way, many which are completed using orthogonality relations. Therefore, since our main concerned is the Now, let X 1, … XK be the irreducible character of G and representations of completely reducible finite groups, we let g 1,…g k be representation of the conjugacy classes of are now set to state our first major result that is G. Then, the K x K matrix where ij-entry is x i (gj) (for all ij Marscke’s theorem without proof. 1< I < = k, 1 < < k) is called the character table of G. Therefore it is usual to number the irreducible Theorem 1 (Marschke’s Theorem) characters and classes of G so that X i = 1 G, the trivial characters and g 1=1 the identity element of G. Also note Let G be a finite group and F a field whose characteristics that the character table, the rows are indexed by the does not divide the order of G (IGI). Then every FG-sub irreducible characters of G and the columns are indexed module U of an FG-module V, ∃ an FG sub module U of by the conjugacy classes or the conjugacy classes’ 1 2 representative. V such that V = U 1⊕ U2 Displayed below is an example of the character table of Definition 5 a finite group. Its column corresponds to the conjugacy classes of G while its row corresponds to the characters χ of the in equivalent irreducible representation of G. The An FG- module V is said to be completely reducible if V = i jth conjugacy class C is indicated by displaying a U ⊕ U ⊕…⊕ U where each U is an irreducible FG – j 1 2 n i representative C C . In the (ij)th entry we put χ (C ). sub module of V. j ∈ j i j Definition 6 (Complete Reducibility) C1 C2 ………….. Cn χi χi(C 1) χi(C 2) ………….. χi(C n) A matrix representation A (X) over F is said to be χ2 χ2(C 1) χ2(C 2) ………….. χ2(C n) completely reducible if A (X) ∼ diag A 1 (x), A 2(x), A i(x). χn χn(C 1) χn(C 2) ………….. χn(C n) Where A i(x) ⇒ i=1,2,… 1 are irreducible representation Scholarly J. Math. Comp. Sci. 10 Remark 1 if i = j δ ij = 0 if i ≠ j Webb (2012) stated that the orthogonality relations provide a check on the accuracy of our calculation and Where δ is the Kronecher delta function also enable us to complete the final row of the character ij table. The facts that the character degrees (dimension) divides order of G and that the sum of squares of the Example: dimension equals order of G also help in this case. Schur’s Lemma is another vital tool for the irreducible Suppose we are given the following part of character modules. table of a group order 12 which have exactly four conjugacy classes. Theorem 2 (Schur’s Lemma) (1) g 1 g 2 g1 g3 g4 Suppose that U and V are irreducible G-module over F. | CG ()g | 12 4 3 4 Then a G –homomorphism θ: V → U is; χ1 1 1 1 1 i. θ is the zero map or else 2 χ2 1 1 Ω Ω ii. θ is an isomorphism 1 1 2 Ω χ3 Ω PROOF e g χ4 f h Let prove the following statements: That is If θ ≠ 0 then θ 2πi is a bijective mapping.
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