2. Examples of Groups 2.1. Some Infinite Abelian Groups. It Is Easy To
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Arxiv:2003.06292V1 [Math.GR] 12 Mar 2020 Eggnrtr N Ignlmti.Tedaoa Arxi Matrix Diagonal the Matrix
ALGORITHMS IN LINEAR ALGEBRAIC GROUPS SUSHIL BHUNIA, AYAN MAHALANOBIS, PRALHAD SHINDE, AND ANUPAM SINGH ABSTRACT. This paper presents some algorithms in linear algebraic groups. These algorithms solve the word problem and compute the spinor norm for orthogonal groups. This gives us an algorithmic definition of the spinor norm. We compute the double coset decompositionwith respect to a Siegel maximal parabolic subgroup, which is important in computing infinite-dimensional representations for some algebraic groups. 1. INTRODUCTION Spinor norm was first defined by Dieudonné and Kneser using Clifford algebras. Wall [21] defined the spinor norm using bilinear forms. These days, to compute the spinor norm, one uses the definition of Wall. In this paper, we develop a new definition of the spinor norm for split and twisted orthogonal groups. Our definition of the spinornorm is rich in the sense, that itis algorithmic in nature. Now one can compute spinor norm using a Gaussian elimination algorithm that we develop in this paper. This paper can be seen as an extension of our earlier work in the book chapter [3], where we described Gaussian elimination algorithms for orthogonal and symplectic groups in the context of public key cryptography. In computational group theory, one always looks for algorithms to solve the word problem. For a group G defined by a set of generators hXi = G, the problem is to write g ∈ G as a word in X: we say that this is the word problem for G (for details, see [18, Section 1.4]). Brooksbank [4] and Costi [10] developed algorithms similar to ours for classical groups over finite fields. -
Classification of Finite Abelian Groups
Math 317 C1 John Sullivan Spring 2003 Classification of Finite Abelian Groups (Notes based on an article by Navarro in the Amer. Math. Monthly, February 2003.) The fundamental theorem of finite abelian groups expresses any such group as a product of cyclic groups: Theorem. Suppose G is a finite abelian group. Then G is (in a unique way) a direct product of cyclic groups of order pk with p prime. Our first step will be a special case of Cauchy’s Theorem, which we will prove later for arbitrary groups: whenever p |G| then G has an element of order p. Theorem (Cauchy). If G is a finite group, and p |G| is a prime, then G has an element of order p (or, equivalently, a subgroup of order p). ∼ Proof when G is abelian. First note that if |G| is prime, then G = Zp and we are done. In general, we work by induction. If G has no nontrivial proper subgroups, it must be a prime cyclic group, the case we’ve already handled. So we can suppose there is a nontrivial subgroup H smaller than G. Either p |H| or p |G/H|. In the first case, by induction, H has an element of order p which is also order p in G so we’re done. In the second case, if ∼ g + H has order p in G/H then |g + H| |g|, so hgi = Zkp for some k, and then kg ∈ G has order p. Note that we write our abelian groups additively. Definition. Given a prime p, a p-group is a group in which every element has order pk for some k. -
Mathematics 310 Examination 1 Answers 1. (10 Points) Let G Be A
Mathematics 310 Examination 1 Answers 1. (10 points) Let G be a group, and let x be an element of G. Finish the following definition: The order of x is ... Answer: . the smallest positive integer n so that xn = e. 2. (10 points) State Lagrange’s Theorem. Answer: If G is a finite group, and H is a subgroup of G, then o(H)|o(G). 3. (10 points) Let ( a 0! ) H = : a, b ∈ Z, ab 6= 0 . 0 b Is H a group with the binary operation of matrix multiplication? Be sure to explain your answer fully. 2 0! 1/2 0 ! Answer: This is not a group. The inverse of the matrix is , which is not 0 2 0 1/2 in H. 4. (20 points) Suppose that G1 and G2 are groups, and φ : G1 → G2 is a homomorphism. (a) Recall that we defined φ(G1) = {φ(g1): g1 ∈ G1}. Show that φ(G1) is a subgroup of G2. −1 (b) Suppose that H2 is a subgroup of G2. Recall that we defined φ (H2) = {g1 ∈ G1 : −1 φ(g1) ∈ H2}. Prove that φ (H2) is a subgroup of G1. Answer:(a) Pick x, y ∈ φ(G1). Then we can write x = φ(a) and y = φ(b), with a, b ∈ G1. Because G1 is closed under the group operation, we know that ab ∈ G1. Because φ is a homomorphism, we know that xy = φ(a)φ(b) = φ(ab), and therefore xy ∈ φ(G1). That shows that φ(G1) is closed under the group operation. -
The General Linear Group
18.704 Gabe Cunningham 2/18/05 [email protected] The General Linear Group Definition: Let F be a field. Then the general linear group GLn(F ) is the group of invert- ible n × n matrices with entries in F under matrix multiplication. It is easy to see that GLn(F ) is, in fact, a group: matrix multiplication is associative; the identity element is In, the n × n matrix with 1’s along the main diagonal and 0’s everywhere else; and the matrices are invertible by choice. It’s not immediately clear whether GLn(F ) has infinitely many elements when F does. However, such is the case. Let a ∈ F , a 6= 0. −1 Then a · In is an invertible n × n matrix with inverse a · In. In fact, the set of all such × matrices forms a subgroup of GLn(F ) that is isomorphic to F = F \{0}. It is clear that if F is a finite field, then GLn(F ) has only finitely many elements. An interesting question to ask is how many elements it has. Before addressing that question fully, let’s look at some examples. ∼ × Example 1: Let n = 1. Then GLn(Fq) = Fq , which has q − 1 elements. a b Example 2: Let n = 2; let M = ( c d ). Then for M to be invertible, it is necessary and sufficient that ad 6= bc. If a, b, c, and d are all nonzero, then we can fix a, b, and c arbitrarily, and d can be anything but a−1bc. This gives us (q − 1)3(q − 2) matrices. -
Unitary Group - Wikipedia
Unitary group - Wikipedia https://en.wikipedia.org/wiki/Unitary_group Unitary group In mathematics, the unitary group of degree n, denoted U( n), is the group of n × n unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group GL( n, C). Hyperorthogonal group is an archaic name for the unitary group, especially over finite fields. For the group of unitary matrices with determinant 1, see Special unitary group. In the simple case n = 1, the group U(1) corresponds to the circle group, consisting of all complex numbers with absolute value 1 under multiplication. All the unitary groups contain copies of this group. The unitary group U( n) is a real Lie group of dimension n2. The Lie algebra of U( n) consists of n × n skew-Hermitian matrices, with the Lie bracket given by the commutator. The general unitary group (also called the group of unitary similitudes ) consists of all matrices A such that A∗A is a nonzero multiple of the identity matrix, and is just the product of the unitary group with the group of all positive multiples of the identity matrix. Contents Properties Topology Related groups 2-out-of-3 property Special unitary and projective unitary groups G-structure: almost Hermitian Generalizations Indefinite forms Finite fields Degree-2 separable algebras Algebraic groups Unitary group of a quadratic module Polynomial invariants Classifying space See also Notes References Properties Since the determinant of a unitary matrix is a complex number with norm 1, the determinant gives a group 1 of 7 2/23/2018, 10:13 AM Unitary group - Wikipedia https://en.wikipedia.org/wiki/Unitary_group homomorphism The kernel of this homomorphism is the set of unitary matrices with determinant 1. -
On Some Generation Methods of Finite Simple Groups
Introduction Preliminaries Special Kind of Generation of Finite Simple Groups The Bibliography On Some Generation Methods of Finite Simple Groups Ayoub B. M. Basheer Department of Mathematical Sciences, North-West University (Mafikeng), P Bag X2046, Mmabatho 2735, South Africa Groups St Andrews 2017 in Birmingham, School of Mathematics, University of Birmingham, United Kingdom 11th of August 2017 Ayoub Basheer, North-West University, South Africa Groups St Andrews 2017 Talk in Birmingham Introduction Preliminaries Special Kind of Generation of Finite Simple Groups The Bibliography Abstract In this talk we consider some methods of generating finite simple groups with the focus on ranks of classes, (p; q; r)-generation and spread (exact) of finite simple groups. We show some examples of results that were established by the author and his supervisor, Professor J. Moori on generations of some finite simple groups. Ayoub Basheer, North-West University, South Africa Groups St Andrews 2017 Talk in Birmingham Introduction Preliminaries Special Kind of Generation of Finite Simple Groups The Bibliography Introduction Generation of finite groups by suitable subsets is of great interest and has many applications to groups and their representations. For example, Di Martino and et al. [39] established a useful connection between generation of groups by conjugate elements and the existence of elements representable by almost cyclic matrices. Their motivation was to study irreducible projective representations of the sporadic simple groups. In view of applications, it is often important to exhibit generating pairs of some special kind, such as generators carrying a geometric meaning, generators of some prescribed order, generators that offer an economical presentation of the group. -
Stabilizers of Lattices in Lie Groups
Journal of Lie Theory Volume 4 (1994) 1{16 C 1994 Heldermann Verlag Stabilizers of Lattices in Lie Groups Richard D. Mosak and Martin Moskowitz Communicated by K. H. Hofmann Abstract. Let G be a connected Lie group with Lie algebra g, containing a lattice Γ. We shall write Aut(G) for the group of all smooth automorphisms of G. If A is a closed subgroup of Aut(G) we denote by StabA(Γ) the stabilizer of Γ n n in A; for example, if G is R , Γ is Z , and A is SL(n;R), then StabA(Γ)=SL(n;Z). The latter is, of course, a lattice in SL(n;R); in this paper we shall investigate, more generally, when StabA(Γ) is a lattice (or a uniform lattice) in A. Introduction Let G be a connected Lie group with Lie algebra g, containing a lattice Γ; so Γ is a discrete subgroup and G=Γ has finite G-invariant measure. We shall write Aut(G) for the group of all smooth automorphisms of G, and M(G) = α Aut(G): mod(α) = 1 for the group of measure-preserving automorphisms off G2 (here mod(α) is thegcommon ratio measure(α(F ))/measure(F ), for any measurable set F G of positive, finite measure). If A is a closed subgroup ⊂ of Aut(G) we denote by StabA(Γ) the stabilizer of Γ in A, in other words α A: α(Γ) = Γ . The main question of this paper is: f 2 g When is StabA(Γ) a lattice (or a uniform lattice) in A? We point out that the thrust of this question is whether the stabilizer StabA(Γ) is cocompact or of cofinite volume, since in any event, in all cases of interest, the stabilizer is discrete (see Prop. -
Orders on Computable Torsion-Free Abelian Groups
Orders on Computable Torsion-Free Abelian Groups Asher M. Kach (Joint Work with Karen Lange and Reed Solomon) University of Chicago 12th Asian Logic Conference Victoria University of Wellington December 2011 Asher M. Kach (U of C) Orders on Computable TFAGs ALC 2011 1 / 24 Outline 1 Classical Algebra Background 2 Computing a Basis 3 Computing an Order With A Basis Without A Basis 4 Open Questions Asher M. Kach (U of C) Orders on Computable TFAGs ALC 2011 2 / 24 Torsion-Free Abelian Groups Remark Disclaimer: Hereout, the word group will always refer to a countable torsion-free abelian group. The words computable group will always refer to a (fixed) computable presentation. Definition A group G = (G : +; 0) is torsion-free if non-zero multiples of non-zero elements are non-zero, i.e., if (8x 2 G)(8n 2 !)[x 6= 0 ^ n 6= 0 =) nx 6= 0] : Asher M. Kach (U of C) Orders on Computable TFAGs ALC 2011 3 / 24 Rank Theorem A countable abelian group is torsion-free if and only if it is a subgroup ! of Q . Definition The rank of a countable torsion-free abelian group G is the least κ cardinal κ such that G is a subgroup of Q . Asher M. Kach (U of C) Orders on Computable TFAGs ALC 2011 4 / 24 Example The subgroup H of Q ⊕ Q (viewed as having generators b1 and b2) b1+b2 generated by b1, b2, and 2 b1+b2 So elements of H look like β1b1 + β2b2 + α 2 for β1; β2; α 2 Z. -
Lie Group and Geometry on the Lie Group SL2(R)
INDIAN INSTITUTE OF TECHNOLOGY KHARAGPUR Lie group and Geometry on the Lie Group SL2(R) PROJECT REPORT – SEMESTER IV MOUSUMI MALICK 2-YEARS MSc(2011-2012) Guided by –Prof.DEBAPRIYA BISWAS Lie group and Geometry on the Lie Group SL2(R) CERTIFICATE This is to certify that the project entitled “Lie group and Geometry on the Lie group SL2(R)” being submitted by Mousumi Malick Roll no.-10MA40017, Department of Mathematics is a survey of some beautiful results in Lie groups and its geometry and this has been carried out under my supervision. Dr. Debapriya Biswas Department of Mathematics Date- Indian Institute of Technology Khargpur 1 Lie group and Geometry on the Lie Group SL2(R) ACKNOWLEDGEMENT I wish to express my gratitude to Dr. Debapriya Biswas for her help and guidance in preparing this project. Thanks are also due to the other professor of this department for their constant encouragement. Date- place-IIT Kharagpur Mousumi Malick 2 Lie group and Geometry on the Lie Group SL2(R) CONTENTS 1.Introduction ................................................................................................... 4 2.Definition of general linear group: ............................................................... 5 3.Definition of a general Lie group:................................................................... 5 4.Definition of group action: ............................................................................. 5 5. Definition of orbit under a group action: ...................................................... 5 6.1.The general linear -
Abstract Algebra
Abstract Algebra Paul Melvin Bryn Mawr College Fall 2011 lecture notes loosely based on Dummit and Foote's text Abstract Algebra (3rd ed) Prerequisite: Linear Algebra (203) 1 Introduction Pure Mathematics Algebra Analysis Foundations (set theory/logic) G eometry & Topology What is Algebra? • Number systems N = f1; 2; 3;::: g \natural numbers" Z = f:::; −1; 0; 1; 2;::: g \integers" Q = ffractionsg \rational numbers" R = fdecimalsg = pts on the line \real numbers" p C = fa + bi j a; b 2 R; i = −1g = pts in the plane \complex nos" k polar form re iθ, where a = r cos θ; b = r sin θ a + bi b r θ a p Note N ⊂ Z ⊂ Q ⊂ R ⊂ C (all proper inclusions, e.g. 2 62 Q; exercise) There are many other important number systems inside C. 2 • Structure \binary operations" + and · associative, commutative, and distributive properties \identity elements" 0 and 1 for + and · resp. 2 solve equations, e.g. 1 ax + bx + c = 0 has two (complex) solutions i p −b ± b2 − 4ac x = 2a 2 2 2 2 x + y = z has infinitely many solutions, even in N (thei \Pythagorian triples": (3,4,5), (5,12,13), . ). n n n 3 x + y = z has no solutions x; y; z 2 N for any fixed n ≥ 3 (Fermat'si Last Theorem, proved in 1995 by Andrew Wiles; we'll give a proof for n = 3 at end of semester). • Abstract systems groups, rings, fields, vector spaces, modules, . A group is a set G with an associative binary operation ∗ which has an identity element e (x ∗ e = x = e ∗ x for all x 2 G) and inverses for each of its elements (8 x 2 G; 9 y 2 G such that x ∗ y = y ∗ x = e). -
SUMMARY of PART II: GROUPS 1. Motivation and Overview 1.1. Why
SUMMARY OF PART II: GROUPS 1. Motivation and overview 1.1. Why study groups? The group concept is truly fundamental not only for algebra, but for mathematics in general, as groups appear in mathematical theories as diverse as number theory (e.g. the solutions of Diophantine equations often form a group), Galois theory (the solutions(=roots) of a polynomial equation may satisfy hidden symmetries which can be put together into a group), geometry (e.g. the isometries of a given geometric space or object form a group) or topology (e.g. an important tool is the fundamental group of a topological space), but also in physics (e.g. the Lorentz group which in concerned with the symmetries of space-time in relativity theory, or the “gauge” symmetry group of the famous standard model). Our task for the course is to understand whole classes of groups—mostly we will concentrate on finite groups which are already very difficult to classify. Our main examples for these shall be the cyclic groups Cn, symmetric groups Sn (on n letters), the alternating groups An and the dihedral groups Dn. The most “fundamental” of these are the symmetric groups, as every finite group can be embedded into a certain finite symmetric group (Cayley’s Theorem). We will be able to classify all groups of order 2p and p2, where p is a prime. We will also find structure results on subgroups which exist for a given such group (Cauchy’s Theorem, Sylow Theorems). Finally, abelian groups can be controlled far easier than general (i.e., possibly non-abelian) groups, and we will give a full classification for all such abelian groups, at least when they can be generated by finitely many elements. -
Bott Periodicity for the Unitary Group
Bott Periodicity for the Unitary Group Carlos Salinas March 7, 2018 Abstract We will present a condensed proof of the Bott Periodicity Theorem for the unitary group U following John Milnor’s classic Morse Theory. There are many documents on the internet which already purport to do this (and do so very well in my estimation), but I nevertheless will attempt to give a summary of the result. Contents 1 The Basics 2 2 Fiber Bundles 3 2.1 First fiber bundle . .4 2.2 Second Fiber Bundle . .5 2.3 Third Fiber Bundle . .5 2.4 Fourth Fiber Bundle . .5 3 Proof of the Periodicity Theorem 6 3.1 The first equivalence . .7 3.2 The second equality . .8 4 The Homotopy Groups of U 8 1 The Basics The original proof of the Periodicity Theorem relies on a deep result of Marston Morse’s calculus of variations, the (Morse) Index Theorem. The proof of this theorem, however, goes beyond the scope of this document, the reader is welcome to read the relevant section from Milnor or indeed Morse’s own paper titled The Index Theorem in the Calculus of Variations. Perhaps the first thing we should set about doing is introducing the main character of our story; this will be the unitary group. The unitary group of degree n (here denoted U(n)) is the set of all unitary matrices; that is, the set of all A ∈ GL(n, C) such that AA∗ = I where A∗ is the conjugate of the transpose of A (conjugate transpose for short).