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A Note on Presentation of General Linear Groups Over a Finite Field
Southeast Asian Bulletin of Mathematics (2019) 43: 217–224 Southeast Asian Bulletin of Mathematics c SEAMS. 2019 A Note on Presentation of General Linear Groups over a Finite Field Swati Maheshwari and R. K. Sharma Department of Mathematics, Indian Institute of Technology Delhi, New Delhi, India Email: [email protected]; [email protected] Received 22 September 2016 Accepted 20 June 2018 Communicated by J.M.P. Balmaceda AMS Mathematics Subject Classification(2000): 20F05, 16U60, 20H25 Abstract. In this article we have given Lie regular generators of linear group GL(2, Fq), n where Fq is a finite field with q = p elements. Using these generators we have obtained presentations of the linear groups GL(2, F2n ) and GL(2, Fpn ) for each positive integer n. Keywords: Lie regular units; General linear group; Presentation of a group; Finite field. 1. Introduction Suppose F is a finite field and GL(n, F) is the general linear the group of n × n invertible matrices and SL(n, F) is special linear group of n × n matrices with determinant 1. We know that GL(n, F) can be written as a semidirect product, GL(n, F)= SL(n, F) oF∗, where F∗ denotes the multiplicative group of F. Let H and K be two groups having presentations H = hX | Ri and K = hY | Si, then a presentation of semidirect product of H and K is given by, −1 H oη K = hX, Y | R,S,xyx = η(y)(x) ∀x ∈ X,y ∈ Y i, where η : K → Aut(H) is a group homomorphism. Now we summarize some literature survey related to the presentation of groups. -
Abelian Varieties with Complex Multiplication and Modular Functions, by Goro Shimura, Princeton Univ
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 36, Number 3, Pages 405{408 S 0273-0979(99)00784-3 Article electronically published on April 27, 1999 Abelian varieties with complex multiplication and modular functions, by Goro Shimura, Princeton Univ. Press, Princeton, NJ, 1998, xiv + 217 pp., $55.00, ISBN 0-691-01656-9 The subject that might be called “explicit class field theory” begins with Kro- necker’s Theorem: every abelian extension of the field of rational numbers Q is a subfield of a cyclotomic field Q(ζn), where ζn is a primitive nth root of 1. In other words, we get all abelian extensions of Q by adjoining all “special values” of e(x)=exp(2πix), i.e., with x Q. Hilbert’s twelfth problem, also called Kronecker’s Jugendtraum, is to do something2 similar for any number field K, i.e., to generate all abelian extensions of K by adjoining special values of suitable special functions. Nowadays we would add that the reciprocity law describing the Galois group of an abelian extension L/K in terms of ideals of K should also be given explicitly. After K = Q, the next case is that of an imaginary quadratic number field K, with the real torus R/Z replaced by an elliptic curve E with complex multiplication. (Kronecker knew what the result should be, although complete proofs were given only later, by Weber and Takagi.) For simplicity, let be the ring of integers in O K, and let A be an -ideal. Regarding A as a lattice in C, we get an elliptic curve O E = C/A with End(E)= ;Ehas complex multiplication, or CM,by .If j=j(A)isthej-invariant ofOE,thenK(j) is the Hilbert class field of K, i.e.,O the maximal abelian unramified extension of K. -
Brauer Groups of Abelian Schemes
ANNALES SCIENTIFIQUES DE L’É.N.S. RAYMOND T. HOOBLER Brauer groups of abelian schemes Annales scientifiques de l’É.N.S. 4e série, tome 5, no 1 (1972), p. 45-70 <http://www.numdam.org/item?id=ASENS_1972_4_5_1_45_0> © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1972, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www. elsevier.com/locate/ansens) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systé- matique est constitutive d’une infraction pénale. Toute copie ou impression de ce fi- chier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Ann. scienL EC. Norm. Sup., 4® serie, t. 5, 1972, p. 45 ^ 70. BRAUER GROUPS OF ABELIAN SCHEMES BY RAYMOND T. HOOBLER 0 Let A be an abelian variety over a field /c. Mumford has given a very beautiful construction of the dual abelian variety in the spirit of Grothen- dieck style algebraic geometry by using the theorem of the square, its corollaries, and cohomology theory. Since the /c-points of Pic^n is H1 (A, G^), it is natural to ask how much of this work carries over to higher cohomology groups where the computations must be made in the etale topology to render them non-trivial. Since H2 (A, Gm) is essentially a torsion group, the representability of the corresponding functor does not have as much geometric interest as for H1 (A, G^). -
Abelian Varieties
Abelian Varieties J.S. Milne Version 2.0 March 16, 2008 These notes are an introduction to the theory of abelian varieties, including the arithmetic of abelian varieties and Faltings’s proof of certain finiteness theorems. The orginal version of the notes was distributed during the teaching of an advanced graduate course. Alas, the notes are still in very rough form. BibTeX information @misc{milneAV, author={Milne, James S.}, title={Abelian Varieties (v2.00)}, year={2008}, note={Available at www.jmilne.org/math/}, pages={166+vi} } v1.10 (July 27, 1998). First version on the web, 110 pages. v2.00 (March 17, 2008). Corrected, revised, and expanded; 172 pages. Available at www.jmilne.org/math/ Please send comments and corrections to me at the address on my web page. The photograph shows the Tasman Glacier, New Zealand. Copyright c 1998, 2008 J.S. Milne. Single paper copies for noncommercial personal use may be made without explicit permis- sion from the copyright holder. Contents Introduction 1 I Abelian Varieties: Geometry 7 1 Definitions; Basic Properties. 7 2 Abelian Varieties over the Complex Numbers. 10 3 Rational Maps Into Abelian Varieties . 15 4 Review of cohomology . 20 5 The Theorem of the Cube. 21 6 Abelian Varieties are Projective . 27 7 Isogenies . 32 8 The Dual Abelian Variety. 34 9 The Dual Exact Sequence. 41 10 Endomorphisms . 42 11 Polarizations and Invertible Sheaves . 53 12 The Etale Cohomology of an Abelian Variety . 54 13 Weil Pairings . 57 14 The Rosati Involution . 61 15 Geometric Finiteness Theorems . 63 16 Families of Abelian Varieties . -
On Balanced Subgroups of the Multiplicative Group
ON BALANCED SUBGROUPS OF THE MULTIPLICATIVE GROUP CARL POMERANCE AND DOUGLAS ULMER In memory of Alf van der Poorten ABSTRACT. A subgroup H of (Z=dZ)× is called balanced if every coset of H is evenly distributed between the lower and upper halves of (Z=dZ)×, i.e., has equal numbers of elements with represen- tatives in (0; d=2) and (d=2; d). This notion has applications to ranks of elliptic curves. We give a simple criterion in terms of characters for a subgroup H to be balanced, and for a fixed integer p, we study the distribution of integers d such that the cyclic subgroup of (Z=dZ)× generated by p is balanced. 1. INTRODUCTION × Let d > 2 be an integer and consider (Z=dZ) , the group of units modulo d. Let Ad be the × first half of (Z=dZ) ; that is, Ad consists of residues with a representative in (0; d=2). Let Bd = × × × (Z=dZ) n Ad be the second half of (Z=dZ) . We say a subgroup H of (Z=dZ) is balanced if × for each g 2 (Z=dZ) we have jgH \ Adj = jgH \ Bdj; that is, each coset of H has equally many members in the first half of (Z=dZ)× as in the second half. Let ' denote Euler’s function, so that φ(d) is the cardinality of (Z=dZ)×. If n and m are coprime integers with m > 0, let ln(m) denote the order of the cyclic subgroup hn mod mi generated by n × in (Z=mZ) (that is, ln(m) is the multiplicative order of n modulo m). -
The Yoneda Ext and Arbitrary Coproducts in Abelian Categories
THE YONEDA EXT AND ARBITRARY COPRODUCTS IN ABELIAN CATEGORIES ALEJANDRO ARGUD´IN MONROY Abstract. There are well known identities involving the Ext bifunctor, co- products, and products in Ab4 abelian categories with enough projectives. Namely, for every such category A, given an object X and a set of ob- n ∼ jects {Ai}i∈I , the following isomorphism can be built ExtA i∈I Ai,X = n n L i∈I ExtA (Ai,X), where Ext is the n-th derived functor of the Hom func- tor.Q The goal of this paper is to show a similar isomorphism for the n-th Yoneda Ext, which is a functor equivalent to Extn that can be defined in more general contexts. The desired isomorphism is constructed explicitly by using colimits, in Ab4 abelian categories with not necessarily enough projec- tives nor injectives, answering a question made by R. Colpi and K R. Fuller in [8]. Furthermore, the isomorphisms constructed are used to characterize Ab4 categories. A dual result is also stated. 1. Introduction The study of extensions is a theory that has developed from multiplicative groups [21, 15], with applications ranging from representations of central simple algebras [4, 13] to topology [10]. In this article we will focus on extensions in an abelian category C. In this context, an extension of an object A by an object C is a short exact sequence 0 → A → M → C → 0 up to equivalence, where two exact sequences are equivalent if there is a morphism arXiv:1904.12182v2 [math.CT] 8 Sep 2020 from one to another with identity morphisms at the ends. -
More on Gauge Theory and Geometric Langlands
More On Gauge Theory And Geometric Langlands Edward Witten School of Natural Sciences, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540 USA Abstract: The geometric Langlands correspondence was described some years ago in terms of S-duality of = 4 super Yang-Mills theory. Some additional matters relevant N to this story are described here. The main goal is to explain directly why an A-brane of a certain simple kind can be an eigenbrane for the action of ’t Hooft operators. To set the stage, we review some facts about Higgs bundles and the Hitchin fibration. We consider only the simplest examples, in which many technical questions can be avoided. arXiv:1506.04293v3 [hep-th] 28 Jul 2017 Contents 1 Introduction 1 2 Compactification And Hitchin’s Moduli Space 2 2.1 Preliminaries 2 2.2 Hitchin’s Equations 4 2.3 MH and the Cotangent Bundle 7 2.4 The Hitchin Fibration 7 2.5 Complete Integrability 8 2.6 The Spectral Curve 10 2.6.1 Basics 10 2.6.2 Abelianization 13 2.6.3 Which Line Bundles Appear? 14 2.6.4 Relation To K-Theory 16 2.6.5 The Unitary Group 17 2.6.6 The Group P SU(N) 18 2.6.7 Spectral Covers For Other Gauge Groups 20 2.7 The Distinguished Section 20 2.7.1 The Case Of SU(N) 20 2.7.2 Section Of The Hitchin Fibration For Any G 22 3 Dual Tori And Hitchin Fibrations 22 3.1 Examples 23 3.2 The Case Of Unitary Groups 25 3.3 Topological Viewpoint 27 3.3.1 Characterization of FSU(N) 29 3.4 The Symplectic Form 30 3.4.1 Comparison To Gauge Theory 33 4 ’t Hooft Operators And Hecke Modifications 34 4.1 Eigenbranes 34 4.2 The Electric -
A STUDY on the ALGEBRAIC STRUCTURE of SL 2(Zpz)
A STUDY ON THE ALGEBRAIC STRUCTURE OF SL2 Z pZ ( ~ ) A Thesis Presented to The Honors Tutorial College Ohio University In Partial Fulfillment of the Requirements for Graduation from the Honors Tutorial College with the degree of Bachelor of Science in Mathematics by Evan North April 2015 Contents 1 Introduction 1 2 Background 5 2.1 Group Theory . 5 2.2 Linear Algebra . 14 2.3 Matrix Group SL2 R Over a Ring . 22 ( ) 3 Conjugacy Classes of Matrix Groups 26 3.1 Order of the Matrix Groups . 26 3.2 Conjugacy Classes of GL2 Fp ....................... 28 3.2.1 Linear Case . .( . .) . 29 3.2.2 First Quadratic Case . 29 3.2.3 Second Quadratic Case . 30 3.2.4 Third Quadratic Case . 31 3.2.5 Classes in SL2 Fp ......................... 33 3.3 Splitting of Classes of(SL)2 Fp ....................... 35 3.4 Results of SL2 Fp ..............................( ) 40 ( ) 2 4 Toward Lifting to SL2 Z p Z 41 4.1 Reduction mod p ...............................( ~ ) 42 4.2 Exploring the Kernel . 43 i 4.3 Generalizing to SL2 Z p Z ........................ 46 ( ~ ) 5 Closing Remarks 48 5.1 Future Work . 48 5.2 Conclusion . 48 1 Introduction Symmetries are one of the most widely-known examples of pure mathematics. Symmetry is when an object can be rotated, flipped, or otherwise transformed in such a way that its appearance remains the same. Basic geometric figures should create familiar examples, take for instance the triangle. Figure 1: The symmetries of a triangle: 3 reflections, 2 rotations. The red lines represent the reflection symmetries, where the trianlge is flipped over, while the arrows represent the rotational symmetry of the triangle. -
Units in Group Rings and Subalgebras of Real Simple Lie Algebras
UNIVERSITY OF TRENTO Faculty of Mathematical, Physical and Natural Sciences Ph.D. Thesis Computational problems in algebra: units in group rings and subalgebras of real simple Lie algebras Advisor: Candidate: Prof. De Graaf Willem Adriaan Faccin Paolo Contents 1 Introduction 3 2 Group Algebras 5 2.1 Classical result about unit group of group algebras . 6 2.1.1 Bass construction . 6 2.1.2 The group of Hoechsmann unit H ............... 7 2.2 Lattices . 8 2.2.1 Ge’s algorithm . 8 2.2.2 Finding a basis of the perp-lattice . 9 2.2.3 The lattice . 11 2.2.4 Pure Lattices . 14 2.3 Toral algebras . 15 2.3.1 Splitting elements in toral algebras . 15 2.3.2 Decomposition via irreducible character of G . 17 2.3.3 Standard generating sets . 17 2.4 Cyclotomic fields Q(ζn) ........................ 18 2.4.1 When n is a prime power . 18 2.4.2 When n is not a prime power . 19 2.4.3 Explicit Construction of Greither ’s Units . 19 2.4.4 Fieker’s program . 24 2.5 Unit groups of orders in toral matrix algebras . 25 2.5.1 A simple toral algebra . 25 2.5.2 Two idempotents . 25 2.5.3 Implementation . 26 2.5.4 The general case . 27 2.6 Units of integral abelian group rings . 27 3 Lie algebras 29 3.0.1 Comment on the notation . 30 3.0.2 Comment on the base field . 31 3.1 Real simple Lie algebras . 31 3.2 Constructing complex semisimple Lie algebras . -
A LOOK at the FUNCTORS TOR and EXT Contents 1. Introduction 1
A LOOK AT THE FUNCTORS TOR AND EXT YEVGENIYA ZHUKOVA Abstract. In this paper I will motivate and define the functors Tor and Ext. I will then discuss a computation of Ext and Tor for finitely generated abelian groups and show their use in the Universal Coefficient Theorem in algebraic topology. Contents 1. Introduction 1 2. Exact Sequence 1 3. Hom 2 4. Tensor Product 3 5. Homology 5 6. Ext 6 7. Tor 9 8. In the Case of Finitely Generated Abelian Groups 9 9. Application: Universal Coefficient Theorem 11 Acknowledgments 12 References 13 1. Introduction In this paper I will be discussing the functors Ext and Tor. In order to accom- plish this I will introduce some key concepts including exact sequences, the tensor product on modules, Hom, chain complexes, and chain homotopies. I will then show how to compute the Ext and Tor groups for finitely generated abelian groups following a paper of J. Michael Boardman, and discuss their import to the Univer- sal Coefficient Theorems in Algebraic Topology. The primary resources referenced include Abstract Algebra by Dummit and Foote [3], Algebraic Topology by Hatcher [5], and An Introduction to the Cohomology of Groups by Peter J. Webb [7]. For this paper I will be assuming a first course in algebra, through the definition of a module over a ring. 2. Exact Sequence Definition 2.1. An exact sequence is a sequence of algebraic structures X; Y; Z and homomorphisms '; between them ' ··· / X / Y / Z / ··· such that Im(') = ker( ) 1 2 YEVGENIYA ZHUKOVA For the purposes of this paper, X; Y; Z will be either abelian groups or R- modules. -
Homological Algebra
HOMOLOGICAL ALGEBRA BRIAN TYRRELL Abstract. In this report we will assemble the pieces of homological algebra needed to explore derived functors from their base in exact se- quences of abelian categories to their realisation as a type of δ-functor, first introduced in 1957 by Grothendieck. We also speak briefly on the typical example of a derived functor, the Ext functor, and note some of its properties. Contents 1. Introduction2 2. Background & Opening Definitions3 2.1. Categories3 2.2. Functors5 2.3. Sequences6 3. Leading to Derived Functors7 4. Chain Homotopies 10 5. Derived Functors 14 5.1. Applications: the Ext functor 20 6. Closing remarks 21 References 22 Date: December 16, 2016. 2 BRIAN TYRRELL 1. Introduction We will begin by defining the notion of a category; Definition 1.1. A category is a triple C = (Ob C; Hom C; ◦) where • Ob C is the class of objects of C. • Hom C is the class of morphisms of C. Furthermore, 8X; Y 2 Ob C we associate a set HomC(X; Y ) - the set of morphisms from X to Y - such that (X; Y ) 6= (Z; U) ) HomC(X; Y ) \ HomC(Z; U) = ;. Finally, we require 8X; Y; Z 2 Ob C the operation ◦ : HomC(Y; Z) × HomC(X; Y ) ! HomC(X; Z)(g; f) 7! g ◦ f to be defined, associative and for all objects the identity morphism must ex- ist, that is, 8X 2 Ob C 91X 2 HomC(X; X) such that 8f 2 HomC(X; Y ); g 2 HomC(Z; X), f ◦ 1X = f and 1X ◦ g = g. -
Fiber Products in Commutative Algebra
Georgia Southern University Digital Commons@Georgia Southern Electronic Theses and Dissertations Graduate Studies, Jack N. Averitt College of Summer 2017 Fiber Products in Commutative Algebra Keller VandeBogert Follow this and additional works at: https://digitalcommons.georgiasouthern.edu/etd Part of the Algebra Commons Recommended Citation VandeBogert, Keller, "Fiber Products in Commutative Algebra" (2017). Electronic Theses and Dissertations. 1608. https://digitalcommons.georgiasouthern.edu/etd/1608 This thesis (open access) is brought to you for free and open access by the Graduate Studies, Jack N. Averitt College of at Digital Commons@Georgia Southern. It has been accepted for inclusion in Electronic Theses and Dissertations by an authorized administrator of Digital Commons@Georgia Southern. For more information, please contact [email protected]. FIBER PRODUCTS IN COMMUTATIVE ALGEBRA by KELLER VANDEBOGERT (Under the Direction of Saeed Nasseh) ABSTRACT The purpose of this thesis is to introduce and illustrate some of the deep connections between commutative and homological algebra. We shall cover some of the fundamental definitions and introduce several important classes of commutative rings. The later chap- ters will consider a particular class of rings, the fiber product, and, among other results, show that any Gorenstein fiber product is precisely a one dimensional hypersurface. It will also be shown that any Noetherian local ring with a (nontrivially) decomposable maximal ideal satisfies the Auslander-Reiten conjecture.