Finite Group Theory

Following the ideas of our previous works math.QA/0008232 (joint with Andruskiewitsch) and math.QA/0101049, we study families of triangular Hopf algebras obtained by twisting finite supergroups by a twist lying entirely in the odd part. These families are parametrized by data (G,V,u,B), where G is a finite group, V its finite dimensional representation, u a central element of G of order 2 acting by -1 on V, and B an element of S^2V. We fix the discrete data G,V,u, and find the set of isomorphism classes of the members of the family as Hopf algebras, in terms of the continuous parameter B. This set is often infinite, which provides examples of nontrivial continuous families of triangular Hopf algebras. The lowest dimension in which such a family occurs is 32, in which case we get 3 families which are dual to the 3 families of pointed Hopf algebras of dimension 32 constructed recently by Grana. Furthermore, we show that if (S^2V)^G=0 then such continuous families are nontrivial not on...

Let $G$ be a finite group and $M(G)$ be the subgroup of $G$ generated by all non-central elements of $G$ that lie in the conjugacy classes of the smallest size. Recently several results have been proved regarding the nilpotency class of $M(G)$ and $F(M(G))$, where $F(M(G))$ denotes the Fitting subgroup of $M(G)$. We prove some conditional results regarding the nilpotency class of $M(G)$.

Using maps due to Ozeki and Broué-Enguehard between graded spaces of invariants for certain finite groups and the algebra of modular forms of even weight we equip these invariants spaces with a differential operator which gives them the structure of a Rankin-Cohen algebra. A direct interpretation of the Rankin-Cohen bracket in terms of transvectant for the group SL(2, C) is given.

A group code structure of a linear code is a description of the code as one-sided or two-sided ideal of a group algebra of a finite group. In these realizations, the group algebra is identified with the ambient space, and the group elements with the coordinates of the ambient space. It is well known that every affine-invariant code of length p m , with p prime, can be realized as an ideal of the group algebra FI, where I is the underlying additive group of the field with p m elements. In this paper we describe all the group code structures of an affine-invariant code of length p m in terms of a family of maps from I to the group of automorphisms of I.

Many results show how restrictions on the values of the irre- ducible characters on the identity element (that is, the degrees of the irreducible characters) of a finite group G, influence the structure of G. In the current article we study groups with restrictions on the values of a nonidentity rational element of the group. More specifically, we show that

An approach to representations of finite groups is presented without recourse to character theory. Considering the group algebra C[G] as an algebra of linear maps on C[G] (by left multiplication), we derive the primitive central idempotents as a simultaneous eigenbasis of the centre, Z(C[G]). We apply this framework to obtain the irreducible representations of a class of finite meta-abelian groups. In particular, we give a general construction of the isomorphism between simple blocks of C[G] and the corresponding matrix algebra where G can be any finite group.

Este texto es una introducción a la teoría de grupos, con énfasis en grupos finitos, el cual fue elaborado para la clase de Teoría de Grupos de la Licenciatura en Matemáticas de la Universidad de Guadalajara. Incluye temas básicos como la definición de grupo, subgrupos, clases laterales, el Teorema de Lagrange, subgrupos normales, grupos cociente, homomorfismos, isomorfismos, automorfismos y los Teoremas de Isomorfia.

Además, incluye secciones especializadas en los siguientes tipos de grupos: grupos cíclicos, donde se examinan los subgrupos de un grupo cíclico y los homomorfismos entre ellos; grupos abelianos, donde se enuncia y aplica el Teorema Fundamental de Grupos Abelianos Finitos;  grupos de permutaciones, donde se definen y estudian las propiedades básicas de los grupos simétricos y alternantes.

Finalmente, las últimas secciones presentan temas sobre acciones de grupos, incluyendo las demostraciones del Teorema Órbita-Estabilizador, el Teorema de Cayley, y el Teorema de Cauchy-Frobenius, y sobre la teoría de Sylow, incluyendo las demostraciones del Teorema de Cauchy, los tres Teoremas de Sylow, y el Teorema Fundamental de Grupos Abelianos Finitos. El texto también incluye dos apéndices: el primero cubre los prerrequisitos necesarios de teoría de números elemental y relaciones de equivalencia, mientras que el segundo incluye una lista de sugerencias de temas para que los estudiantes de un curso elaboren un proyecto final.

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This book is primarily intended for undergraduate students in Mathematics, pursuing courses that feature abstract algebra. All aspects of abstract algebra that you would expect to encounter in an undergraduate programme of study are covered, including ring theory, group theory and the beginnings of Galois theory. Unusually for an abstract algebra text, five chapters on linear algebra are also included, making the text a self-contained introduction to undergraduate algebra. The book will be of use throughout your undergraduate studies, and beyond. Very little is assumed by way of previous study.  It is assumed only that you are familiar with basic topics in algebra, such as complex numbers, matrices, and solving systems of linear equations. From these beginnings, you will be led step by step into the study of various algebraic structures, including groups, vector spaces and rings, and structure-preserving mappings between such objects such as homomorphisms of groups and linear transformations of vector spaces. Important theorems and results are presented and discussed at each stage, with proofs that will enable you to understand the logical development of the subject. Mathematical proof is an aspect of the subject which many students find challenging, and care has been taken to ensure that proofs are easy to follow with all steps clearly explained. In the last six chapters, you will be introduced to more novel topics that go beyond the typical undergraduate curriculum. These include exterior algebras, matrix groups and their associated Lie algebras, normed real algebras and Clifford algebras. Any of these topics could form the basis for a final year undergraduate dissertation, or provide a gentle introduction to graduate study in algebra. Throughout the text, the emphasis is upon presenting topics in an accessible manner, with worked examples which will help you to build a firm understanding of each new concept as it is introduced. Each chapter concludes with a comprehensive set of exercises designed to test and consolidate your understanding of the ideas presented. Solutions to the exercises are given at the end of the book. The book is based on lectures given by Michael Butler at the University of Bolton in the U.K. between 1994 and 2019. These were attended by students from an exceptionally wide range of backgrounds. The lecture notes that this book grew from were refined in the light of teaching and in response to feedback from students, over many deliveries of the material. Dr Butler has extensive experience of teaching abstract algebra to undergraduates, and his passion for teaching the subject has shaped this book.

CONTENTS:
1 The Group Axioms and Examples
2 Subgroups and Group Homomorphisms
3 Vector Spaces
4 Subspaces and Linear Transformations
5 The Basis for a Vector Space
6 Eigenvalues and Eigenvectors
7 Inner Product Spaces
8 Cosets and Quotient Groups
9 Group Actions
10 Simple Groups
11 Soluble Groups
12 Fields and their Extensions
13 The Galois Group
14 The Ring Axioms and Examples
15 Subrings, Ideals and Ring Homomorphisms
16 Quotient Rings
17 Integral Domains and Fields
18 Finite Fields
19 Factorisation in an Integral Domain
20 Vector Spaces with Products
21 The Exterior Algebra of a Vector Space
22 Lie Algebras
23 Matrix Groups and their Tangent Spaces
24 Normed Real Algebras
25 Tensor Products and Clifford Algebras.

In this article, the character tables of some finite groups are constructed. The Mathematical knowledge needed to do such constructions is developed within the first three chapters. The modular theory approach to representation theory was adopted in this work; hence strong emphasis was placed on modular representation. Use was also made of the computer system GAP (see ) in this work.

For a finite groupG, #Cent(G) denotes the number of centralizers of its elements. A groupG is called n-centralizer if #Cent(G) =n, and primitiven-centralizer if $\# Cent(G){\text{ = \# }}Cent{\text{(}}\frac{G}{{Z(G)}}){\text{ = }}n$ . In this paper we compute the number of distinct centralizers of some finite groups and investigate the structure of finite groups with exactly six distinct centralizers. We prove that ifG is a 6-centralizer group then % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca% WGhbaabaGaamOwaiaacIcacaWGhbGaaiykaaaacaqGGaGaeyyrIaKa% aeiiaGqaciaa-readaWgaaWcbaGaa8hoaaqabaGccaGGSaGaaeiiai% aa-feadaWgaaWcbaGaa8hnaaqabaGccaGGSaGaaeiiaiaabQfadaWg% aaWcbaGaaeOmaaqabaGccaqGGaGaey41aqRaaeiiaiaabQfadaWgaa% WcbaGaaeOmaaqabaGccaqGGaGaey41aqRaaeiiaiaabQfadaWgaaWc% baGaaeOmaaqabaGccaqGGaGaae4BaiaabkhacaqGGaGaaeOwamaaBa% aaleaacaqGYaaabeaakiaabccacqGHxdaTcaqGGaGaaeOwamaaBaaa% leaacaqGYaaabeaakiaabccacqGHxdaTcaqGGaGaaeOwamaaBaaale% aacaqGYaaabeaakiaabccacqGHxdaTcaqGGaGaaeOwamaaBaaaleaa% caqGYaaabeaaaaa!62C4!\[\frac{G}{{Z(G)}}{\text{ }} \cong {\text{ }}D_8 ,{\text{ }}A_4 ,{\text{ Z}}_{\text{2}} {\text{ }} \times {\text{ Z}}_{\text{2}} {\text{ }} \times {\text{ Z}}_{\text{2}} {\text{ or Z}}_{\text{2}} {\text{ }} \times {\text{ Z}}_{\text{2}} {\text{ }} \times {\text{ Z}}_{\text{2}} {\text{ }} \times {\text{ Z}}_{\text{2}} \] .

For a finite group G, let Cent(G) denote the set of centralizers of single elements of G and #Cent(G) = |Cent(G)|. G is called an n-centralizer group if #Cent(G) = n, and a primitive n-centralizer group if #Cent(G) = #Cent(G/Z(G)) = n. In this paper, we compute #Cent(G) for some finite groups G and prove that, for any positive integer n ≠ 2, 3, there exists a finite group G with #Cent(G) = n, which is a question raised by Belcastro and Sherman [2]. We investigate the structure of finite groups G with #Cent(G) = 6 and prove that, if G is a primitive 6-centralizer group, then G/Z(G) ≅ A 4, the alternating group on four letters. Also, we prove that, if G/Z(G) ≅ A 4, then #Cent(G) = 6 or 8, and construct a group G with G/Z(G) ≅ A 4 and #Cent(G) = 8.

La transitività della normalità in teoria dei gruppi è strettamente correlata alla proprietà dei sottogruppi di essere pronormali. Ammettere solo sottogruppi pronormali è sufficiente per godere della proprietà T.
Partendo dall'intransitività della normalità è iniziato lo studio di quei gruppi in cui ogni sottogruppo subnormale è anche normale, detti T-gruppi. La struttura dei T-gruppi risolubili finiti è stata descritta completamente da W. Gaschutz, mentre D. J. S. Robinson ha studiato i T-gruppi risolubili nel caso infinito.
La definizione di pronormalità è stata introdotta da P. Hall e i primi risultati sono presenti nell'articolo di J. S. Rose "Finite soluble groups with pronormal system normalizers" del 1967.
Nel corso della tesi vengono esposte le principali proprietà della pronormalità in teoria dei gruppi e a tal fine si sono studiate preliminarmente la classe dei T-gruppi, dei gruppi risolubili, nilpotenti e localmente nilpotenti.
Si conclude con un recente risultato che mostra come la pronormalità è una proprietà gruppale non locale contabilmente riconoscibili.

To any finite group Γ⊂Sp(V) of automorphisms of a symplectic vector space V we associate a new multi-parameter deformation, Hκ of the algebra ℂ[V]#Γ, smash product of Γ with the polynomial algebra on V. The parameter κ runs over points of ℙ r , where r=number of conjugacy classes of symplectic reflections in $Γ$. The algebra Hκ, called a symplectic reflection algebra, is related to the coordinate ring of a Poisson deformation of the quotient singularity V/Γ. This leads to a symplectic analogue of McKay correspondence, which is most complete in case of wreath-products. If Γ is the Weyl group of a root system in a vector space ? and V=?⊕?*, then the algebras Hκ are certain ‘rational’ degenerations of the double affine Hecke algebra introduced earlier by Cherednik. Let $Γ=S n , the Weyl group of ?=?? n . We construct a 1-parameter deformation of the Harish-Chandra homomorphism from ?(?)?, the algebra of invariant polynomial differential operators on ?? n , to the algebra of S n -invariant differential operators with rational coefficients on the space ℂ n of diagonal matrices. The second order Laplacian on ? goes, under the deformed homomorphism, to the Calogero-Moser differential operator on ℂ n , with rational potential. Our crucial idea is to reinterpret the deformed Harish-Chandra homomorphism as a homomorphism: ?(?)? ↠ spherical subalgebra in Hκ, where Hκ is the symplectic reflection algebra associated to the group Γ=S n . This way, the deformed Harish-Chandra homomorphism becomes nothing but a description of the spherical subalgebra in terms of ‘quantum’ Hamiltonian reduction. In the ‘classical’ limit κ→∞, our construction gives an isomorphism between the spherical subalgebra in H∞ and the coordinate ring of the Calogero-Moser space. We prove that all simple H∞-modules have dimension n!, and are parametrised by points of the Calogero-Moser space. The family of these modules forms a distinguished vector bundle on the Calogero-Moser space, whose fibers carry the regular representation of S n . Moreover, we prove that the algebra $H∞ is isomorphic to the endomorphism algebra of that vector bundle.

The purpose of this article is to give a proof of the Orbifold Theorem announced by Thurston in late 1981: If $O$ is a compact, connected, orientable, irreducible and topologically atoroidal 3-orbifold with non-empty ramification locus, then $O$ is geometric. As a corollary, any smooth orientation preserving non-free finite group action on $S^3$ is conjugate to an orthogonal action.

Representation theory of groups: the first part concern the representation theory of finite groups (with also character theory) and the second part the discussion is focused on the Lie Groups/Algebras.

The topics are:

Introduction

Basic notions
One dimensional representations (abelian groups)
Character of a representation                                             
Reduction and induction of representations

Non finite groups ( Algebraic sets , Lie algebra)
Solvability and (semi-)simplicity 
Root systems and Dynkin diagrams
Universal enveloping Lie algebra

At the end many exercises are solved

Let \pi(G) denote the set of prime divisors of the order of a finite group G. The prime graph of G is the graph with vertex set \pi(G) with edges {p,q} if and only if there exists an element of order pq in G. In this paper, we prove that a graph is isomorphic to the prime graph of a solvable group if and only if its complement is 3-colorable and triangle free. We then introduce the idea of a minimal prime graph. We prove that there exists an infinite class of solvable groups whose prime graphs are minimal. We prove the 3k-conjecture on prime divisors in element orders for solvable groups with minimal prime graphs, and we show that solvable groups whose prime graphs are minimal have Fitting length 3 or 4.

Estas son las notas del curso impartido en el Centro Universitario de Ciencias Excactas e Ingenierías de la Universidad de Guadalajara del 4 al 8 de abril de 2011. La teoría de representaciones de grupos es un área del álgebra abstracta que estudia algunas conexiones entre el álgebra lineal y la teoría de grupos. En este texto estudiaremos las representaciones …complejas de grupos finitos.

We associate a graph Γ G to a non locally cyclic group G (called the non-cyclic graph of G) as follows: take G\Cyc(G) as vertex set, where Cyc(G) = {x ∈ G | x, y is cyclic for all y ∈ G}, and join two vertices if they do not generate a cyclic subgroup. We study the properties of this graph and we establish some graph theoretical properties (such as regularity) of this graph in terms of the group ones. We prove that the clique number of Γ G is finite if and only if Γ G has no infinite clique. We prove that if G is a finite nilpotent group and H is a group with Γ G ∼ = Γ H and |Cyc(G)| = |Cyc(H)| = 1, then H is a finite nilpotent group. We give some examples of groups G whose non-cyclic graphs are "unique", i.e., if Γ G ∼ = Γ H for some group H, then G ∼ = H. In view of these examples, we conjecture that every finite non-abelian simple group has a unique non-cyclic graph. Also we give some examples of finite non-cyclic groups G with the property that if Γ G ∼ = Γ H for some group H, then |G| = |H|. These suggest the question whether the latter property holds for all finite non-cyclic groups.

Emmy Noether’s many articles around the time that Felix Klein and David Hilbert were arranging her invitation to Göttingen include a short but brilliant note on invariants of finite groups highlighting her creativity and perspicacity in algebra. Contrary to the idea that Noether abandoned Paul Gordan’s style of mathematics for Hilbert’s, this note shows her combining them in a way she continued throughout her mature abstract algebra.

Proof. All the non-abelian subgroups of G are G and Q 8 . Furthermore, G = Q 8 and Q 8 = Z (Q 8 ), the center of Q 8 , which are normal in G. Hence D(G) = G, and G is non-abelian. 2 Example 1.3. As Aut(Q 8 ) ∼ = S 4 and D 8 S 4 , we have the semidirect product G = [Q 8 ]D 8 . Then G is a 2-group of order 2 6 and D(G) < G.

A Latin square of order n is an n × n array of n symbols, in which each symbol occurs exactly once in each row and column. A transversal is a set of n entries, one selected from each row and each column of a Latin square of order n such that no two entries contain the same symbol. Define T (n) to be the maximum number of transversals over all Latin squares of order n. We show that b n ≤ T (n) ≤ c n √ n n! for n ≥ 5, where b ≈ 1.719 and c ≈ 0.614. A corollary of this result is an upper bound on the number of placements of n non-attacking queens on an n × n toroidal chess board. Some divisibility properties of the number of transversals in Latin squares based on finite groups are established. We also provide data from a computer enumeration of transversals in all Latin squares of order at most 9, all groups of order at most 23 and all possible turn-squares of order 14.

Smooth and symplectic symmetries of an infinite family of distinct exotic K3 surfaces are studied, and comparison with the corresponding symmetries of the standard K3 is made. The action on the K3 lattice induced by a smooth finite group action is shown to be strongly restricted, and as a result, nonsmoothability of actions induced by a holomorphic automorphism of a prime order ≥ 7 is proved and nonexistence of smooth actions by several K3 groups is established (included among which is the binary tetrahedral group T24 which has the smallest order). Concerning symplectic symmetries, the fixed-point set structure of a symplectic cyclic action of a prime order ≥ 5 is explicitly determined, provided that the action is homologically nontrivial.

The second author introduced notions of weak permutablity and commutativity between groups and proved the finiteness of a group generated by two weakly permutable finite subgroups. Two groups H, K weakly commute provided there exists a bijection f : H → K which fixes the identity and such that h commutes with its image h f for all h ∈ H . The present paper gives support to conjectures about the nilpotency of groups generated by two weakly commuting finite abelian groups H, K .

Un elemento finito lineal con sección transversal constante puede adoptar cualquier orientación en el plano y sus extremos o nodos lo ligan al resto de los elementos. La energ a cinética (T ) y potencial (V ) de un elemento elástico dinámico son el basamento en la implementación del principio de Hamilton para la definición de un elemento finito. La definición de la energía cinética y potencial es el primer paso para la formulación variacional preliminar a la enunciación por elementos finitos que se utiliza para resolver, dígase, los problemas de mecanismos que se mueven en el plano utilizando la ecuación de Hamilton. El objetivo general consistió en definir la ecuación del movimiento de un elemento finito lineal plano elástico dinámico utilizando la ecuación de Hamilton, a partir de la lagrangiana (T –V ) obtenida con el uso de un polinomio de quinto y uno de primer grados, con ocho grados de libertad, cuatro en cada nodo, que representaron las deformaciones: axial (u(x)), transversal (w(x)), pendiente ((dw(x)/dx)) y curvatura ((d2w(x)/dx2)). La deformación debido al cizalleo transversal, insignificante comparado con la deformación flexional y la axial, la inercia rotatoria y las fuerzas friccionales en las uniones,fueron desestimadas con el fin de producir un elemento amigo. Los objetivos específicos fueron producir: (a) la matriz de masa de traslación [MD], (b) la matriz giroscópica de traslación [AD], (c) la matriz de rigidez total de traslación [KD], y (d) el vector de deformación (S). Como resultado se forjó la ecuación del movimiento de un elemento finito lineal plano elástico dinámico [MD]( ¨ S) − 2¨_[AD]( ˙S ) + {[K] − _˙2[MD] − ¨_[AD]}(S) = (Q) . Se concluyó que la ecuación obtenida variacionalmente con la aplicación del principio de Hamilton es un modelo cuyo procedimiento puede ser utilizado cuando se requiera aumentar el número de grados de libertad del modelo.

Um grupo cristalográfico tridimensional é um subgrupo do grupo de simetrias de um reticulado. Todo o grupo cristalográfico pontual (isto é, um grupo finito gerado por translações de  R^3 ) é um grupo cristalográfico cujo reticulado contém um ponto fixo.
Este trabalho consiste na classificação dos grupos cristalográficos pontuais tendo como objectivo primordial a elaboração de um texto que em termos linguísticos e matemáticos esteja acessível a um aluno do último ano da licenciatura em Ensino da Matemática da Universidade de Aveiro.


A three-dimensional crystallographic group is a simmetry subgroup of a lattice. Every crystallographic point group (which is a finite group generated by translations on R^3 ) is a crystallographic group whose the lattice has one fixed point.
This work aims at classifying crystallographic point group in order to present a readable text to undergraduate students in their last year of the Maths Teaching degree of the University of Aveiro.

The present paper deals with a maximal subgroup of the Thompson group, namely the group 2 1+8 + · A9 := G. We compute its conjugacy classes using the coset analysis method, its inertia factor groups and Fischer matrices, which are required for the computations of the character table of G by means of Clifford-Fischer Theory.

Finite group extensions offer a natural language to quantum computing. In a nutshell, one roughly describes the action of a quantum computer as consisting of two finite groups of gates: error gates from the general Pauli group P and stabilizing gates within an extension group C. In this paper one explores the nice adequacy between group theoretical concepts such as commutators, normal subgroups, group of automorphisms, short exact sequences, wreath products... and the coherent quantum computational primitives. The structure of the single qubit and two-qubit Clifford groups is analyzed in detail. As a byproduct, one discovers that M 20 , the smallest perfect group for which the commutator subgroup departs from the set of commutators, underlies quantum coherence of the two-qubit system. One recovers similar results by looking at the automorphisms of a complete set of mutually unbiased bases.

The partial group algebra of a group G over a field K, denoted by Kpar(G), is the algebra whose representations correspond to the partial representations of G over K-vector spaces. In this paper we study the structure of the partial group algebra Kpar(G), where G is a finite group. In particular, given two finite abelian groups G1 and G2, we prove that if the characteristic of K is zero, then Kpar(G1) is isomorphic to Kpar(G2) if and only if G1 is isomorphic to G2.