Computing the Character Table of a Finite Group
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Frattini Subgroups and -Central Groups
Pacific Journal of Mathematics FRATTINI SUBGROUPS AND 8-CENTRAL GROUPS HOMER FRANKLIN BECHTELL,JR. Vol. 18, No. 1 March 1966 PACIFIC JOURNAL OF MATHEMATICS Vol. 18, No. 1, 1966 FRATTINI SUBGROUPS AND φ-CENTRAL GROUPS HOMER BECHTELL 0-central groups are introduced as a step In the direction of determining sufficiency conditions for a group to be the Frattini subgroup of some unite p-gronp and the related exten- sion problem. The notion of Φ-centrality arises by uniting the concept of an E-group with the generalized central series of Kaloujnine. An E-group is defined as a finite group G such that Φ(N) ^ Φ(G) for each subgroup N ^ G. If Sίf is a group of automorphisms of a group N, N has an i^-central series ι a N = No > Nt > > Nr = 1 if x~x e N3- for all x e Nj-lf all a a 6 £%f, x the image of x under the automorphism a e 3ίf y i = 0,l, •••, r-1. Denote the automorphism group induced OR Φ(G) by trans- formation of elements of an £rgroup G by 3ίf. Then Φ{£ίf) ~ JP'iΦiG)), J^iβiG)) the inner automorphism group of Φ(G). Furthermore if G is nilpotent9 then each subgroup N ^ Φ(G), N invariant under 3ίf \ possess an J^-central series. A class of niipotent groups N is defined as ^-central provided that N possesses at least one niipotent group of automorphisms ££'' Φ 1 such that Φ{βίf} — ,J^(N) and N possesses an J^-central series. Several theorems develop results about (^-central groups and the associated ^^-central series analogous to those between niipotent groups and their associated central series. -
NOTES on FINITE GROUP REPRESENTATIONS in Fall 2020, I
NOTES ON FINITE GROUP REPRESENTATIONS CHARLES REZK In Fall 2020, I taught an undergraduate course on abstract algebra. I chose to spend two weeks on the theory of complex representations of finite groups. I covered the basic concepts, leading to the classification of representations by characters. I also briefly addressed a few more advanced topics, notably induced representations and Frobenius divisibility. I'm making the lectures and these associated notes for this material publicly available. The material here is standard, and is mainly based on Steinberg, Representation theory of finite groups, Ch 2-4, whose notation I will mostly follow. I also used Serre, Linear representations of finite groups, Ch 1-3.1 1. Group representations Given a vector space V over a field F , we write GL(V ) for the group of bijective linear maps T : V ! V . n n When V = F we can write GLn(F ) = GL(F ), and identify the group with the group of invertible n × n matrices. A representation of a group G is a homomorphism of groups φ: G ! GL(V ) for some representation choice of vector space V . I'll usually write φg 2 GL(V ) for the value of φ on g 2 G. n When V = F , so we have a homomorphism φ: G ! GLn(F ), we call it a matrix representation. matrix representation The choice of field F matters. For now, we will look exclusively at the case of F = C, i.e., representations in complex vector spaces. Remark. Since R ⊆ C is a subfield, GLn(R) is a subgroup of GLn(C). -
Molecular Symmetry
Molecular Symmetry Symmetry helps us understand molecular structure, some chemical properties, and characteristics of physical properties (spectroscopy) – used with group theory to predict vibrational spectra for the identification of molecular shape, and as a tool for understanding electronic structure and bonding. Symmetrical : implies the species possesses a number of indistinguishable configurations. 1 Group Theory : mathematical treatment of symmetry. symmetry operation – an operation performed on an object which leaves it in a configuration that is indistinguishable from, and superimposable on, the original configuration. symmetry elements – the points, lines, or planes to which a symmetry operation is carried out. Element Operation Symbol Identity Identity E Symmetry plane Reflection in the plane σ Inversion center Inversion of a point x,y,z to -x,-y,-z i Proper axis Rotation by (360/n)° Cn 1. Rotation by (360/n)° Improper axis S 2. Reflection in plane perpendicular to rotation axis n Proper axes of rotation (C n) Rotation with respect to a line (axis of rotation). •Cn is a rotation of (360/n)°. •C2 = 180° rotation, C 3 = 120° rotation, C 4 = 90° rotation, C 5 = 72° rotation, C 6 = 60° rotation… •Each rotation brings you to an indistinguishable state from the original. However, rotation by 90° about the same axis does not give back the identical molecule. XeF 4 is square planar. Therefore H 2O does NOT possess It has four different C 2 axes. a C 4 symmetry axis. A C 4 axis out of the page is called the principle axis because it has the largest n . By convention, the principle axis is in the z-direction 2 3 Reflection through a planes of symmetry (mirror plane) If reflection of all parts of a molecule through a plane produced an indistinguishable configuration, the symmetry element is called a mirror plane or plane of symmetry . -
GROUP REPRESENTATIONS and CHARACTER THEORY Contents 1
GROUP REPRESENTATIONS AND CHARACTER THEORY DAVID KANG Abstract. In this paper, we provide an introduction to the representation theory of finite groups. We begin by defining representations, G-linear maps, and other essential concepts before moving quickly towards initial results on irreducibility and Schur's Lemma. We then consider characters, class func- tions, and show that the character of a representation uniquely determines it up to isomorphism. Orthogonality relations are introduced shortly afterwards. Finally, we construct the character tables for a few familiar groups. Contents 1. Introduction 1 2. Preliminaries 1 3. Group Representations 2 4. Maschke's Theorem and Complete Reducibility 4 5. Schur's Lemma and Decomposition 5 6. Character Theory 7 7. Character Tables for S4 and Z3 12 Acknowledgments 13 References 14 1. Introduction The primary motivation for the study of group representations is to simplify the study of groups. Representation theory offers a powerful approach to the study of groups because it reduces many group theoretic problems to basic linear algebra calculations. To this end, we assume that the reader is already quite familiar with linear algebra and has had some exposure to group theory. With this said, we begin with a preliminary section on group theory. 2. Preliminaries Definition 2.1. A group is a set G with a binary operation satisfying (1) 8 g; h; i 2 G; (gh)i = g(hi)(associativity) (2) 9 1 2 G such that 1g = g1 = g; 8g 2 G (identity) (3) 8 g 2 G; 9 g−1 such that gg−1 = g−1g = 1 (inverses) Definition 2.2. -
Quasi P Or Not Quasi P? That Is the Question
Rose-Hulman Undergraduate Mathematics Journal Volume 3 Issue 2 Article 2 Quasi p or not Quasi p? That is the Question Ben Harwood Northern Kentucky University, [email protected] Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Recommended Citation Harwood, Ben (2002) "Quasi p or not Quasi p? That is the Question," Rose-Hulman Undergraduate Mathematics Journal: Vol. 3 : Iss. 2 , Article 2. Available at: https://scholar.rose-hulman.edu/rhumj/vol3/iss2/2 Quasi p- or not quasi p-? That is the Question.* By Ben Harwood Department of Mathematics and Computer Science Northern Kentucky University Highland Heights, KY 41099 e-mail: [email protected] Section Zero: Introduction The question might not be as profound as Shakespeare’s, but nevertheless, it is interesting. Because few people seem to be aware of quasi p-groups, we will begin with a bit of history and a definition; and then we will determine for each group of order less than 24 (and a few others) whether the group is a quasi p-group for some prime p or not. This paper is a prequel to [Hwd]. In [Hwd] we prove that (Z3 £Z3)oZ2 and Z5 o Z4 are quasi 2-groups. Those proofs now form a portion of Proposition (12.1) It should also be noted that [Hwd] may also be found in this journal. Section One: Why should we be interested in quasi p-groups? In a 1957 paper titled Coverings of algebraic curves [Abh2], Abhyankar conjectured that the algebraic fundamental group of the affine line over an algebraically closed field k of prime characteristic p is the set of quasi p-groups, where by the algebraic fundamental group of the affine line he meant the family of all Galois groups Gal(L=k(X)) as L varies over all finite normal extensions of k(X) the function field of the affine line such that no point of the line is ramified in L, and where by a quasi p-group he meant a finite group that is generated by all of its p-Sylow subgroups. -
Combinatorial Group Theory
Combinatorial Group Theory Charles F. Miller III 7 March, 2004 Abstract An early version of these notes was prepared for use by the participants in the Workshop on Algebra, Geometry and Topology held at the Australian National University, 22 January to 9 February, 1996. They have subsequently been updated and expanded many times for use by students in the subject 620-421 Combinatorial Group Theory at the University of Melbourne. Copyright 1996-2004 by C. F. Miller III. Contents 1 Preliminaries 3 1.1 About groups . 3 1.2 About fundamental groups and covering spaces . 5 2 Free groups and presentations 11 2.1 Free groups . 12 2.2 Presentations by generators and relations . 16 2.3 Dehn’s fundamental problems . 19 2.4 Homomorphisms . 20 2.5 Presentations and fundamental groups . 22 2.6 Tietze transformations . 24 2.7 Extraction principles . 27 3 Construction of new groups 30 3.1 Direct products . 30 3.2 Free products . 32 3.3 Free products with amalgamation . 36 3.4 HNN extensions . 43 3.5 HNN related to amalgams . 48 3.6 Semi-direct products and wreath products . 50 4 Properties, embeddings and examples 53 4.1 Countable groups embed in 2-generator groups . 53 4.2 Non-finite presentability of subgroups . 56 4.3 Hopfian and residually finite groups . 58 4.4 Local and poly properties . 61 4.5 Finitely presented coherent by cyclic groups . 63 1 5 Subgroup Theory 68 5.1 Subgroups of Free Groups . 68 5.1.1 The general case . 68 5.1.2 Finitely generated subgroups of free groups . -
Representations, Character Tables, and One Application of Symmetry Chapter 4
Representations, Character Tables, and One Application of Symmetry Chapter 4 Friday, October 2, 2015 Matrices and Matrix Multiplication A matrix is an array of numbers, Aij column columns matrix row matrix -1 4 3 1 1234 rows -8 -1 7 2 2141 3 To multiply two matrices, add the products, element by element, of each row of the first matrix with each column in the second matrix: 12 12 (1×1)+(2×3) (1×2)+(2×4) 710 × 34 34 = (3×1)+(4×3) (3×2)+(4×4) = 15 22 100 1 1 0-10× 2 = -2 002 3 6 Transformation Matrices Each symmetry operation can be represented by a 3×3 matrix that shows how the operation transforms a set of x, y, and z coordinates y Let’s consider C2h {E, C2, i, σh}: x transformation matrix C2 x’ = -x -1 0 0 x’ -1 0 0 x -x y’ = -y 0-10 y’ ==0-10 y -y z’ = z 001 z’ 001 z z new transformation old new in terms coordinates ==matrix coordinates of old transformation i matrix x’ = -x -1 0 0 x’ -1 0 0 x -x y’ = -y 0-10 y’ ==0-10 y -y z’ = -z 00-1 z’ 00-1z -z Representations of Groups The set of four transformation matrices forms a matrix representation of the C2h point group. 100 -1 0 0 -1 0 0 100 E: 010 C2: 0-10 i: 0-10 σh: 010 001 001 00-1 00-1 These matrices combine in the same way as the operations, e.g., -1 0 0 -1 0 0 100 C2 × C2 = 0-10 0-10= 010= E 001001 001 The sum of the numbers along each matrix diagonal (the character) gives a shorthand version of the matrix representation, called Γ: C2h EC2 i σh Γ 3-1-31 Γ (gamma) is a reducible representation b/c it can be further simplified. -
Lecture Notes
i Applications of Group Theory to the Physics of Solids M. S. Dresselhaus 8.510J 6.734J SPRING 2002 ii Contents 1 Basic Mathematical Background 1 1.1 De¯nition of a Group . 1 1.2 Simple Example of a Group . 2 1.3 Basic De¯nitions . 4 1.4 Rearrangement Theorem . 6 1.5 Cosets . 6 1.6 Conjugation and Class . 8 1.6.1 Self-Conjugate Subgroups . 9 1.7 Factor Groups . 10 1.8 Selected Problems . 11 2 Representation Theory 15 2.1 Important De¯nitions . 15 2.2 Matrices . 17 2.3 Irreducible Representations . 18 2.4 The Unitarity of Representations . 20 2.5 Schur's Lemma (Part I) . 23 2.6 Schur's Lemma (Part 2) . 24 2.7 Wonderful Orthogonality Theorem . 27 2.8 Representations and Vector Spaces . 30 2.9 Suggested Problems . 31 3 Character of a Representation 33 3.1 De¯nition of Character . 33 3.2 Characters and Class . 34 3.3 Wonderful Orthogonality Theorem for Character . 36 3.4 Reducible Representations . 38 iii iv CONTENTS 3.5 The Number of Irreducible Representations . 40 3.6 Second Orthogonality Relation for Characters . 41 3.7 Regular Representation . 43 3.8 Setting up Character Tables . 47 3.9 Symmetry Notation . 51 3.10 Selected Problems . 73 4 Basis Functions 75 4.1 Symmetry Operations and Basis Functions . 75 4.2 Basis Functions for Irreducible Representations . 77 ^(¡n) 4.3 Projection Operators Pkl . 82 ^(¡n) 4.4 Derivation of Pk` . 83 4.5 Projection Operations on an Arbitrary Function . 84 4.6 Linear Combinations for 3 Equivalent Atoms . -
2-Elementary Subgroups of the Space Cremona Group 3
2-elementary subgroups of the space Cremona group Yuri Prokhorov ∗ Abstract We give a sharp bound for orders of elementary abelian 2-groups of bira- tional automorphisms of rationally connected threefolds. 1 Introduction Throughoutthis paper we work over k, an algebraically closed field of characteristic 0. Recall that the Cremona group Crn(k) is the group of birational transformations n of the projective space Pk. We are interested in finite subgroupsof Crn(k). For n = 2 these subgroups are classified basically (see [DI09] and references therein) but for n ≥ 3 the situation becomes much more complicated. There are only a few, very specific classification results (see e.g. [Pr12], [Pr11], [Pr13c]). Let p be a prime number. A group G is said to be p-elementary abelian of rank r if G ≃ (Z/pZ)r. In this case we denote r(G) := r. A. Beauville [Be07] obtained a sharp bound for the rank of p-elementary abelian subgroups of Cr2(k). Theorem 1.1 ([Be07]). Let G ⊂ Cr2(k) be a 2-elementary abelian subgroup. Then r(G) ≤ 4. Moreover, this bound is sharp and such groups G with r(G)= 4 are clas- sified up to conjugacy in Cr2(k). The author [Pr11] was able to get a similar bound for p-elementary abelian sub- groups of Cr3(k) which is sharp for p ≥ 17. arXiv:1308.5628v1 [math.AG] 26 Aug 2013 In this paper we improve this result in the case p = 2. We study 2-elementary abelian subgroups acting on rationally connected threefolds. In particular, we obtain Steklov Mathematical Institute, 8 Gubkina str., Moscow 119991 · Laboratory of Algebraic Geom- etry, SU-HSE, 7 Vavilova str., Moscow 117312 e-mail: [email protected] ∗ I acknowledge partial supports by RFBR grants No. -
The Q-Conjugacy Character Table of Dihedral Groups
ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS { N. 39{2018 (89{96) 89 THE Q-CONJUGACY CHARACTER TABLE OF DIHEDRAL GROUPS H. Shabani A. R. Ashrafi E. Haghi Department of Pure Mathematics Faculty of Mathematical Sciences University of Kashan P.O. Box 87317-53153 Kashan I.R. Iran M. Ghorbani∗ Department of Pure Mathematics Faculty of Sciences Shahid Rajaee Teacher Training University P.O. Box 16785-136 Tehran I.R. Iran [email protected] Abstract. In a seminal paper published in 1998, Shinsaku Fujita introduced the concept of Q-conjugacy character table of a finite group. He applied this notion to solve some problems in combinatorial chemistry. In this paper, the Q-conjugacy character table of dihedral groups is computed in general. As a consequence, Q- ∼ ∼ conjugacy character table of molecules with point group symmetries D = C = Dih , ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼3 3v∼ 6 D = D = C = Dih , D = C = Dih , D = D = D = C = Dih , 4 ∼ 2d ∼ 4v ∼ 8 5 ∼5v 10 ∼3d ∼3h 6 ∼6v 12 D2 = C2v = Z2 × Z2 = Dih2, D4d = Dih16, D5d = D5h = Dih20, D6d = Dih24 are computed, where Z2 denotes a cyclic group of order 2 and Dihn is the dihedral group of even order n. Keywords: Q-conjugacy character table, dihedral group, conjugacy class. 1. Introduction A representation of a group G is a homomorphism from G into the group of invertible operators of a vector space V . In this case, we can interpret each element of G as an invertible linear transformation V −! V . If n = dimV and fix a basis for V then we can construct an isomorphism between the set of all \invertible linear transformation V −! V " and the set of all \invertible n × n matrices". -
Irreducible Representations and Character Tables
MIT OpenCourseWare http://ocw.mit.edu 5.04 Principles of Inorganic Chemistry II �� Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 5.04, Principles of Inorganic Chemistry II Prof. Daniel G. Nocera Lecture 3: Irreducible Representations and Character Tables Similarity transformations yield irreducible representations, Γi, which lead to the useful tool in group theory – the character table. The general strategy for determining Γi is as follows: A, B and C are matrix representations of symmetry operations of an arbitrary basis set (i.e., elements on which symmetry operations are performed). There is some similarity transform operator v such that A’ = v –1 ⋅ A ⋅ v B’ = v –1 ⋅ B ⋅ v C’ = v –1 ⋅ C ⋅ v where v uniquely produces block-diagonalized matrices, which are matrices possessing square arrays along the diagonal and zeros outside the blocks ⎡ A1 ⎤ ⎡B1 ⎤ ⎡C1 ⎤ ′ = ⎢ ⎥ ′ = ⎢ ⎥ ′ = ⎢ ⎥ A ⎢ A 2 ⎥ B ⎢ B2 ⎥ C ⎢ C 2 ⎥ ⎢ A ⎥ ⎢ B ⎥ ⎢ C ⎥ ⎣ 3 ⎦ ⎣ 3 ⎦ ⎣ 3 ⎦ Matrices A, B, and C are reducible. Sub-matrices Ai, Bi and Ci obey the same multiplication properties as A, B and C. If application of the similarity transform does not further block-diagonalize A’, B’ and C’, then the blocks are irreducible representations. The character is the sum of the diagonal elements of Γi. 2 As an example, let’s continue with our exemplary group: E, C3, C3 , σv, σv’, σv” by defining an arbitrary basis … a triangle v A v'' v' C B The basis set is described by the triangles vertices, points A, B and C. -
A Point Group Character Tables
A Point Group Character Tables Appendix A contains Point Group Character (Tables A.1–A.34) to be used throughout the chapters of this book. Pedagogic material to assist the reader in the use of these character tables can be found in Chap. 3. The Schoenflies symmetry (Sect. 3.9) and Hermann–Mauguin notations (Sect. 3.10) for the point groups are also discussed in Chap. 3. Some of the more novel listings in this appendix are the groups with five- fold symmetry C5, C5h, C5v, D5, D5d, D5h, I, Ih. The cubic point group Oh in Table A.31 lists basis functions for all the irreducible representations of Oh and uses the standard solid state physics notation for the irreducible representations. Table A.1. Character table for group C1 (triclinic) C1 (1) E A 1 Table A.2. Character table for group Ci = S2 (triclinic) S2 (1) Ei 2 2 2 x ,y ,z ,xy,xz,yz Rx,Ry,Rz Ag 11 x, y, z Au 1 −1 Table A.3. Character table for group C1h = S1 (monoclinic) C1h(m) Eσh 2 2 2 x ,y ,z ,xy Rz,x,y A 11 xz, yz Rx,Ry,z A 1 −1 480 A Point Group Character Tables Table A.4. Character table for group C2 (monoclinic) C2 (2) EC2 2 2 2 x ,y ,z ,xy Rz,z A 11 (x, y) xz, yz B 1 −1 (Rx,Ry) Table A.5. Character table for group C2v (orthorhombic) C2v (2mm) EC2 σv σv 2 2 2 x ,y ,z z A1 1111 xy Rz A2 11−1 −1 xz Ry,x B1 1 −11−1 yz Rx,y B2 1 −1 −11 Table A.6.