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Group Homomorphisms
1-17-2018 Group Homomorphisms Here are the operation tables for two groups of order 4: · 1 a a2 + 0 1 2 1 1 a a2 0 0 1 2 a a a2 1 1 1 2 0 a2 a2 1 a 2 2 0 1 There is an obvious sense in which these two groups are “the same”: You can get the second table from the first by replacing 0 with 1, 1 with a, and 2 with a2. When are two groups the same? You might think of saying that two groups are the same if you can get one group’s table from the other by substitution, as above. However, there are problems with this. In the first place, it might be very difficult to check — imagine having to write down a multiplication table for a group of order 256! In the second place, it’s not clear what a “multiplication table” is if a group is infinite. One way to implement a substitution is to use a function. In a sense, a function is a thing which “substitutes” its output for its input. I’ll define what it means for two groups to be “the same” by using certain kinds of functions between groups. These functions are called group homomorphisms; a special kind of homomorphism, called an isomorphism, will be used to define “sameness” for groups. Definition. Let G and H be groups. A homomorphism from G to H is a function f : G → H such that f(x · y)= f(x) · f(y) forall x,y ∈ G. -
Categories of Sets with a Group Action
Categories of sets with a group action Bachelor Thesis of Joris Weimar under supervision of Professor S.J. Edixhoven Mathematisch Instituut, Universiteit Leiden Leiden, 13 June 2008 Contents 1 Introduction 1 1.1 Abstract . .1 1.2 Working method . .1 1.2.1 Notation . .1 2 Categories 3 2.1 Basics . .3 2.1.1 Functors . .4 2.1.2 Natural transformations . .5 2.2 Categorical constructions . .6 2.2.1 Products and coproducts . .6 2.2.2 Fibered products and fibered coproducts . .9 3 An equivalence of categories 13 3.1 G-sets . 13 3.2 Covering spaces . 15 3.2.1 The fundamental group . 15 3.2.2 Covering spaces and the homotopy lifting property . 16 3.2.3 Induced homomorphisms . 18 3.2.4 Classifying covering spaces through the fundamental group . 19 3.3 The equivalence . 24 3.3.1 The functors . 25 4 Applications and examples 31 4.1 Automorphisms and recovering the fundamental group . 31 4.2 The Seifert-van Kampen theorem . 32 4.2.1 The categories C1, C2, and πP -Set ................... 33 4.2.2 The functors . 34 4.2.3 Example . 36 Bibliography 38 Index 40 iii 1 Introduction 1.1 Abstract In the 40s, Mac Lane and Eilenberg introduced categories. Although by some referred to as abstract nonsense, the idea of categories allows one to talk about mathematical objects and their relationions in a general setting. Its origins lie in the field of algebraic topology, one of the topics that will be explored in this thesis. First, a concise introduction to categories will be given. -
Självständiga Arbeten I Matematik
SJÄLVSTÄNDIGA ARBETEN I MATEMATIK MATEMATISKA INSTITUTIONEN, STOCKHOLMS UNIVERSITET Geometric interpretation of non-associative composition algebras av Fredrik Cumlin 2020 - No K13 MATEMATISKA INSTITUTIONEN, STOCKHOLMS UNIVERSITET, 106 91 STOCKHOLM Geometric interpretation of non-associative composition algebras Fredrik Cumlin Självständigt arbete i matematik 15 högskolepoäng, grundnivå Handledare: Wushi Goldring 2020 Abstract This paper aims to discuss the connection between non-associative composition algebra and geometry. It will first recall the notion of an algebra, and investigate the properties of an algebra together with a com- position norm. The composition norm will induce a law on the algebra, which is stated as the composition law. This law is then used to derive the multiplication and conjugation laws, where the last is also known as convolution. These laws are then used to prove Hurwitz’s celebrated the- orem concerning the different finite composition algebras. More properties of composition algebras will be covered, in order to look at the structure of the quaternions H and octonions O. The famous Fano plane will be the finishing touch of the relationship between the standard orthogonal vectors which construct the octonions. Lastly, the notion of invertible maps in relation to invertible loops will be covered, to later show the connection between 8 dimensional rotations − and multiplication of unit octonions. 2 Contents 1 Algebra 4 1.1 The multiplication laws . .6 1.2 The conjugation laws . .7 1.3 Dickson double . .8 1.4 Hurwitz’s theorem . 11 2 Properties of composition algebras 14 2.1 The left-, right- and bi-multiplication maps . 16 2.2 Basic properties of quaternions and octonions . -
LECTURE 12: LIE GROUPS and THEIR LIE ALGEBRAS 1. Lie
LECTURE 12: LIE GROUPS AND THEIR LIE ALGEBRAS 1. Lie groups Definition 1.1. A Lie group G is a smooth manifold equipped with a group structure so that the group multiplication µ : G × G ! G; (g1; g2) 7! g1 · g2 is a smooth map. Example. Here are some basic examples: • Rn, considered as a group under addition. • R∗ = R − f0g, considered as a group under multiplication. • S1, Considered as a group under multiplication. • Linear Lie groups GL(n; R), SL(n; R), O(n) etc. • If M and N are Lie groups, so is their product M × N. Remarks. (1) (Hilbert's 5th problem, [Gleason and Montgomery-Zippin, 1950's]) Any topological group whose underlying space is a topological manifold is a Lie group. (2) Not every smooth manifold admits a Lie group structure. For example, the only spheres that admit a Lie group structure are S0, S1 and S3; among all the compact 2 dimensional surfaces the only one that admits a Lie group structure is T 2 = S1 × S1. (3) Here are two simple topological constraints for a manifold to be a Lie group: • If G is a Lie group, then TG is a trivial bundle. n { Proof: We identify TeG = R . The vector bundle isomorphism is given by φ : G × TeG ! T G; φ(x; ξ) = (x; dLx(ξ)) • If G is a Lie group, then π1(G) is an abelian group. { Proof: Suppose α1, α2 2 π1(G). Define α : [0; 1] × [0; 1] ! G by α(t1; t2) = α1(t1) · α2(t2). Then along the bottom edge followed by the right edge we have the composition α1 ◦ α2, where ◦ is the product of loops in the fundamental group, while along the left edge followed by the top edge we get α2 ◦ α1. -
Math 412. Adventure Sheet on Quotient Groups
Math 412. Adventure sheet on Quotient Groups Fix an arbitrary group (G; ◦). DEFINITION: A subgroup N of G is normal if for all g 2 G, the left and right N-cosets gN and Ng are the same subsets of G. NOTATION: If H ⊆ G is any subgroup, then G=H denotes the set of left cosets of H in G. Its elements are sets denoted gH where g 2 G. The cardinality of G=H is called the index of H in G. DEFINITION/THEOREM 8.13: Let N be a normal subgroup of G. Then there is a well-defined binary operation on the set G=N defined as follows: G=N × G=N ! G=N g1N ? g2N = (g1 ◦ g2)N making G=N into a group. We call this the quotient group “G modulo N”. A. WARMUP: Define the sign map: Sn ! {±1g σ 7! 1 if σ is even; σ 7! −1 if σ is odd. (1) Prove that sign map is a group homomorphism. (2) Use the sign map to give a different proof that An is a normal subgroup of Sn for all n. (3) Describe the An-cosets of Sn. Make a table to describe the quotient group structure Sn=An. What is the identity element? B. OPERATIONS ON COSETS: Let (G; ◦) be a group and let N ⊆ G be a normal subgroup. (1) Take arbitrary ng 2 Ng. Prove that there exists n0 2 N such that ng = gn0. (2) Take any x 2 g1N and any y 2 g2N. Prove that xy 2 g1g2N. -
Right-Angled Artin Groups in the C Diffeomorphism Group of the Real Line
Right-angled Artin groups in the C∞ diffeomorphism group of the real line HYUNGRYUL BAIK, SANG-HYUN KIM, AND THOMAS KOBERDA Abstract. We prove that every right-angled Artin group embeds into the C∞ diffeomorphism group of the real line. As a corollary, we show every limit group, and more generally every countable residually RAAG ∞ group, embeds into the C diffeomorphism group of the real line. 1. Introduction The right-angled Artin group on a finite simplicial graph Γ is the following group presentation: A(Γ) = hV (Γ) | [vi, vj] = 1 if and only if {vi, vj}∈ E(Γ)i. Here, V (Γ) and E(Γ) denote the vertex set and the edge set of Γ, respec- tively. ∞ For a smooth oriented manifold X, we let Diff+ (X) denote the group of orientation preserving C∞ diffeomorphisms on X. A group G is said to embed into another group H if there is an injective group homormophism G → H. Our main result is the following. ∞ R Theorem 1. Every right-angled Artin group embeds into Diff+ ( ). Recall that a finitely generated group G is a limit group (or a fully resid- ually free group) if for each finite set F ⊂ G, there exists a homomorphism φF from G to a nonabelian free group such that φF is an injection when restricted to F . The class of limit groups fits into a larger class of groups, which we call residually RAAG groups. A group G is in this class if for each arXiv:1404.5559v4 [math.GT] 16 Feb 2015 1 6= g ∈ G, there exists a graph Γg and a homomorphism φg : G → A(Γg) such that φg(g) 6= 1. -
MA4E0 Lie Groups
MA4E0 Lie Groups December 5, 2012 Contents 0 Topological Groups 1 1 Manifolds 2 2 Lie Groups 3 3 The Lie Algebra 6 4 The Tangent space 8 5 The Lie Algebra of a Lie Group 10 6 Integrating Vector fields and the exponential map 13 7 Lie Subgroups 17 8 Continuous homomorphisms are smooth 22 9 Distributions and Frobenius Theorem 23 10 Lie Group topics 24 These notes are based on the 2012 MA4E0 Lie Groups course, taught by John Rawnsley, typeset by Matthew Egginton. No guarantee is given that they are accurate or applicable, but hopefully they will assist your study. Please report any errors, factual or typographical, to [email protected] i MA4E0 Lie Groups Lecture Notes Autumn 2012 0 Topological Groups Let G be a group and then write the group structure in terms of maps; multiplication becomes m : G × G ! G −1 defined by m(g1; g2) = g1g2 and inversion becomes i : G ! G defined by i(g) = g . If we suppose that there is a topology on G as a set given by a subset T ⊂ P (G) with the usual rules. We give G × G the product topology. Then we require that m and i are continuous maps. Then G with this topology is a topological group Examples include 1. G any group with the discrete topology 2. Rn with the Euclidean topology and m(x; y) = x + y and i(x) = −x 1 2 2 2 3. S = f(x; y) 2 R : x + y = 1g = fz 2 C : jzj = 1g with m(z1; z2) = z1z2 and i(z) =z ¯. -
Bi-Lagrangian Structures and Teichmüller Theory Brice Loustau, Andrew Sanders
Bi-Lagrangian structures and Teichmüller theory Brice Loustau, Andrew Sanders To cite this version: Brice Loustau, Andrew Sanders. Bi-Lagrangian structures and Teichmüller theory. 2017. hal- 01579284v3 HAL Id: hal-01579284 https://hal.archives-ouvertes.fr/hal-01579284v3 Preprint submitted on 23 Aug 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Bi-Lagrangian structures and Teichmüller theory Brice Loustau∗ and Andrew Sanders† Abstract In this paper, we review or introduce several differential structures on manifolds in the general setting of real and complex differential geometry, and apply this study to Teichmüller theory. We focus on bi-Lagrangian i.e. para-Kähler structures, which consist of a symplectic form and a pair of transverse Lagrangian foliations. We prove in particular that the complexification of a real-analytic Kähler manifold has a natural complex bi-Lagrangian structure. We then specialize to moduli spaces of geometric structures on closed surfaces, and show that old and new geometric features of these spaces are formal consequences of the general theory. We also gain clarity on several well-known results of Teichmüller theory by deriving them from pure differential geometric machinery. -
What Does a Lie Algebra Know About a Lie Group?
WHAT DOES A LIE ALGEBRA KNOW ABOUT A LIE GROUP? HOLLY MANDEL Abstract. We define Lie groups and Lie algebras and show how invariant vector fields on a Lie group form a Lie algebra. We prove that this corre- spondence respects natural maps and discuss conditions under which it is a bijection. Finally, we introduce the exponential map and use it to write the Lie group operation as a function on its Lie algebra. Contents 1. Introduction 1 2. Lie Groups and Lie Algebras 2 3. Invariant Vector Fields 3 4. Induced Homomorphisms 5 5. A General Exponential Map 8 6. The Campbell-Baker-Hausdorff Formula 9 Acknowledgments 10 References 10 1. Introduction Lie groups provide a mathematical description of many naturally-occuring sym- metries. Though they take a variety of shapes, Lie groups are closely linked to linear objects called Lie algebras. In fact, there is a direct correspondence between these two concepts: simply-connected Lie groups are isomorphic exactly when their Lie algebras are isomorphic, and every finite-dimensional real or complex Lie algebra occurs as the Lie algebra of a simply-connected Lie group. To put it another way, a simply-connected Lie group is completely characterized by the small collection n2(n−1) of scalars that determine its Lie bracket, no more than 2 numbers for an n-dimensional Lie group. In this paper, we introduce the basic Lie group-Lie algebra correspondence. We first define the concepts of a Lie group and a Lie algebra and demonstrate how a certain set of functions on a Lie group has a natural Lie algebra structure. -
Groups and Categories
\chap04" 2009/2/27 i i page 65 i i 4 GROUPS AND CATEGORIES This chapter is devoted to some of the various connections between groups and categories. If you already know the basic group theory covered here, then this will give you some insight into the categorical constructions we have learned so far; and if you do not know it yet, then you will learn it now as an application of category theory. We will focus on three different aspects of the relationship between categories and groups: 1. groups in a category, 2. the category of groups, 3. groups as categories. 4.1 Groups in a category As we have already seen, the notion of a group arises as an abstraction of the automorphisms of an object. In a specific, concrete case, a group G may thus consist of certain arrows g : X ! X for some object X in a category C, G ⊆ HomC(X; X) But the abstract group concept can also be described directly as an object in a category, equipped with a certain structure. This more subtle notion of a \group in a category" also proves to be quite useful. Let C be a category with finite products. The notion of a group in C essentially generalizes the usual notion of a group in Sets. Definition 4.1. A group in C consists of objects and arrows as so: m i G × G - G G 6 u 1 i i i i \chap04" 2009/2/27 i i page 66 66 GROUPSANDCATEGORIES i i satisfying the following conditions: 1. -
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. -
Bi-Lagrangian Structures and Teichmüller Theory
Bi-Lagrangian structures and Teichmüller theory Brice Loustau∗ and Andrew Sanders† Abstract This paper has two purposes: the first is to study several structures on manifolds in the general setting of real and complex differential geometry; the second is to apply this study to Teichmüller theory. We primarily focus on bi-Lagrangian structures, which are the data of a symplectic structure and a pair of transverse Lagrangian foliations, and are equivalent to para-Kähler structures. First we carefully study real and complex bi-Lagrangian structures and discuss other closely related structures and their interrelationships. Next we prove the existence of a canonical complex bi-Lagrangian structure in the complexification of any real-analytic Kähler manifold and showcase its properties. We later use this bi-Lagrangian structure to construct a natural almost hyper-Hermitian structure. We then specialize our study to moduli spaces of geometric structures on closed surfaces, which tend to have a rich symplectic structure. We show that some of the recognized geometric features of these moduli spaces are formal consequences of the general theory, while revealing other new geometric features. We also gain clarity on several well-known results of Teichmüller theory by deriving them from pure differential geometric machinery. Key words and phrases: Bi-Lagrangian · para-Kähler · complex geometry · symplectic geometry of moduli spaces · Teichmüller theory · quasi-Fuchsian · hyper-Kähler 2000 Mathematics Subject Classification: Primary: 53C15; Secondary: 30F60 · 32Q15 · 53B05 · 53B35 · 53C26 · 53D05 · 53D30 · 57M50 ∗Rutgers University - Newark, Department of Mathematics and Computer Science. Newark, NJ 07105 USA. E-mail: [email protected] †Heidelberg University, Mathematisches Institut.