THE KERNEL of MONOID MORPHISMS Introduction
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Chapter 4. Homomorphisms and Isomorphisms of Groups
Chapter 4. Homomorphisms and Isomorphisms of Groups 4.1 Note: We recall the following terminology. Let X and Y be sets. When we say that f is a function or a map from X to Y , written f : X ! Y , we mean that for every x 2 X there exists a unique corresponding element y = f(x) 2 Y . The set X is called the domain of f and the range or image of f is the set Image(f) = f(X) = f(x) x 2 X . For a set A ⊆ X, the image of A under f is the set f(A) = f(a) a 2 A and for a set −1 B ⊆ Y , the inverse image of B under f is the set f (B) = x 2 X f(x) 2 B . For a function f : X ! Y , we say f is one-to-one (written 1 : 1) or injective when for every y 2 Y there exists at most one x 2 X such that y = f(x), we say f is onto or surjective when for every y 2 Y there exists at least one x 2 X such that y = f(x), and we say f is invertible or bijective when f is 1:1 and onto, that is for every y 2 Y there exists a unique x 2 X such that y = f(x). When f is invertible, the inverse of f is the function f −1 : Y ! X defined by f −1(y) = x () y = f(x). For f : X ! Y and g : Y ! Z, the composite g ◦ f : X ! Z is given by (g ◦ f)(x) = g(f(x)). -
Complete Objects in Categories
Complete objects in categories James Richard Andrew Gray February 22, 2021 Abstract We introduce the notions of proto-complete, complete, complete˚ and strong-complete objects in pointed categories. We show under mild condi- tions on a pointed exact protomodular category that every proto-complete (respectively complete) object is the product of an abelian proto-complete (respectively complete) object and a strong-complete object. This to- gether with the observation that the trivial group is the only abelian complete group recovers a theorem of Baer classifying complete groups. In addition we generalize several theorems about groups (subgroups) with trivial center (respectively, centralizer), and provide a categorical explana- tion behind why the derivation algebra of a perfect Lie algebra with trivial center and the automorphism group of a non-abelian (characteristically) simple group are strong-complete. 1 Introduction Recall that Carmichael [19] called a group G complete if it has trivial cen- ter and each automorphism is inner. For each group G there is a canonical homomorphism cG from G to AutpGq, the automorphism group of G. This ho- momorphism assigns to each g in G the inner automorphism which sends each x in G to gxg´1. It can be readily seen that a group G is complete if and only if cG is an isomorphism. Baer [1] showed that a group G is complete if and only if every normal monomorphism with domain G is a split monomorphism. We call an object in a pointed category complete if it satisfies this latter condi- arXiv:2102.09834v1 [math.CT] 19 Feb 2021 tion. -
Combinatorial Operads from Monoids Samuele Giraudo
Combinatorial operads from monoids Samuele Giraudo To cite this version: Samuele Giraudo. Combinatorial operads from monoids. Journal of Algebraic Combinatorics, Springer Verlag, 2015, 41 (2), pp.493-538. 10.1007/s10801-014-0543-4. hal-01236471 HAL Id: hal-01236471 https://hal.archives-ouvertes.fr/hal-01236471 Submitted on 1 Dec 2015 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. COMBINATORIAL OPERADS FROM MONOIDS SAMUELE GIRAUDO Abstract. We introduce a functorial construction which, from a monoid, produces a set- operad. We obtain new (symmetric or not) operads as suboperads or quotients of the operads obtained from usual monoids such as the additive and multiplicative monoids of integers and cyclic monoids. They involve various familiar combinatorial objects: endo- functions, parking functions, packed words, permutations, planar rooted trees, trees with a fixed arity, Schröder trees, Motzkin words, integer compositions, directed animals, and segmented integer compositions. We also recover some already known (symmetric or not) operads: the magmatic operad, the associative commutative operad, the diassociative op- erad, and the triassociative operad. We provide presentations by generators and relations of all constructed nonsymmetric operads. Contents Introduction 1 1. Syntax trees and operads3 1.1. -
Category of G-Groups and Its Spectral Category
Communications in Algebra ISSN: 0092-7872 (Print) 1532-4125 (Online) Journal homepage: http://www.tandfonline.com/loi/lagb20 Category of G-Groups and its Spectral Category María José Arroyo Paniagua & Alberto Facchini To cite this article: María José Arroyo Paniagua & Alberto Facchini (2017) Category of G-Groups and its Spectral Category, Communications in Algebra, 45:4, 1696-1710, DOI: 10.1080/00927872.2016.1222409 To link to this article: http://dx.doi.org/10.1080/00927872.2016.1222409 Accepted author version posted online: 07 Oct 2016. Published online: 07 Oct 2016. Submit your article to this journal Article views: 12 View related articles View Crossmark data Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=lagb20 Download by: [UNAM Ciudad Universitaria] Date: 29 November 2016, At: 17:29 COMMUNICATIONS IN ALGEBRA® 2017, VOL. 45, NO. 4, 1696–1710 http://dx.doi.org/10.1080/00927872.2016.1222409 Category of G-Groups and its Spectral Category María José Arroyo Paniaguaa and Alberto Facchinib aDepartamento de Matemáticas, División de Ciencias Básicas e Ingeniería, Universidad Autónoma Metropolitana, Unidad Iztapalapa, Mexico, D. F., México; bDipartimento di Matematica, Università di Padova, Padova, Italy ABSTRACT ARTICLE HISTORY Let G be a group. We analyse some aspects of the category G-Grp of G-groups. Received 15 April 2016 In particular, we show that a construction similar to the construction of the Revised 22 July 2016 spectral category, due to Gabriel and Oberst, and its dual, due to the second Communicated by T. Albu. author, is possible for the category G-Grp. -
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. -
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. -
Kernel and Image
Math 217 Worksheet 1 February: x3.1 Professor Karen E Smith (c)2015 UM Math Dept licensed under a Creative Commons By-NC-SA 4.0 International License. T Definitions: Given a linear transformation V ! W between vector spaces, we have 1. The source or domain of T is V ; 2. The target of T is W ; 3. The image of T is the subset of the target f~y 2 W j ~y = T (~x) for some x 2 Vg: 4. The kernel of T is the subset of the source f~v 2 V such that T (~v) = ~0g. Put differently, the kernel is the pre-image of ~0. Advice to the new mathematicians from an old one: In encountering new definitions and concepts, n m please keep in mind concrete examples you already know|in this case, think about V as R and W as R the first time through. How does the notion of a linear transformation become more concrete in this special case? Think about modeling your future understanding on this case, but be aware that there are other important examples and there are important differences (a linear map is not \a matrix" unless *source and target* are both \coordinate spaces" of column vectors). The goal is to become comfortable with the abstract idea of a vector space which embodies many n features of R but encompasses many other kinds of set-ups. A. For each linear transformation below, determine the source, target, image and kernel. 2 3 x1 3 (a) T : R ! R such that T (4x25) = x1 + x2 + x3. -
Representations of Certain Semidirect Product Groups* H
JOURNAL OF FUNCTIONAL ANALYSIS 19, 339-372 (1975) Representations of Certain Semidirect Product Groups* JOSEPH A. WOLF Department of Mathematics, University of California, Berkeley, California 94720 Communicated by the Editovs Received April I, 1974 We study a class of semidirect product groups G = N . U where N is a generalized Heisenberg group and U is a generalized indefinite unitary group. This class contains the Poincare group and the parabolic subgroups of the simple Lie groups of real rank 1. The unitary representations of G and (in the unimodular cases) the Plancherel formula for G are written out. The problem of computing Mackey obstructions is completely avoided by realizing the Fock representations of N on certain U-invariant holomorphic cohomology spaces. 1. INTRODUCTION AND SUMMARY In this paper we write out the irreducible unitary representations for a type of semidirect product group that includes the Poincare group and the parabolic subgroups of simple Lie groups of real rank 1. If F is one of the four finite dimensional real division algebras, the corresponding groups in question are the Np,Q,F . {U(p, 4; F) xF+}, where FP~*: right vector space F”+Q with hermitian form h(x, y) = i xiyi - y xipi ) 1 lJ+1 N p,p,F: Im F + F”q* with (w. , ~o)(w, 4 = (w, + 7.0+ Im h(z, ,4, z. + 4, U(p, Q; F): unitary group of Fp,q, F+: subgroup of the group generated by F-scalar multiplication on Fp>*. * Research partially supported by N.S.F. Grant GP-16651. 339 Copyright 0 1975 by Academic Press, Inc. -
Derived Functors and Homological Dimension (Pdf)
DERIVED FUNCTORS AND HOMOLOGICAL DIMENSION George Torres Math 221 Abstract. This paper overviews the basic notions of abelian categories, exact functors, and chain complexes. It will use these concepts to define derived functors, prove their existence, and demon- strate their relationship to homological dimension. I affirm my awareness of the standards of the Harvard College Honor Code. Date: December 15, 2015. 1 2 DERIVED FUNCTORS AND HOMOLOGICAL DIMENSION 1. Abelian Categories and Homology The concept of an abelian category will be necessary for discussing ideas on homological algebra. Loosely speaking, an abelian cagetory is a type of category that behaves like modules (R-mod) or abelian groups (Ab). We must first define a few types of morphisms that such a category must have. Definition 1.1. A morphism f : X ! Y in a category C is a zero morphism if: • for any A 2 C and any g; h : A ! X, fg = fh • for any B 2 C and any g; h : Y ! B, gf = hf We denote a zero morphism as 0XY (or sometimes just 0 if the context is sufficient). Definition 1.2. A morphism f : X ! Y is a monomorphism if it is left cancellative. That is, for all g; h : Z ! X, we have fg = fh ) g = h. An epimorphism is a morphism if it is right cancellative. The zero morphism is a generalization of the zero map on rings, or the identity homomorphism on groups. Monomorphisms and epimorphisms are generalizations of injective and surjective homomorphisms (though these definitions don't always coincide). It can be shown that a morphism is an isomorphism iff it is epic and monic. -
Math 120 Homework 3 Solutions
Math 120 Homework 3 Solutions Xiaoyu He, with edits by Prof. Church April 21, 2018 [Note from Prof. Church: solutions to starred problems may not include all details or all portions of the question.] 1.3.1* Let σ be the permutation 1 7! 3; 2 7! 4; 3 7! 5; 4 7! 2; 5 7! 1 and let τ be the permutation 1 7! 5; 2 7! 3; 3 7! 2; 4 7! 4; 5 7! 1. Find the cycle decompositions of each of the following permutations: σ; τ; σ2; στ; τσ; τ 2σ. The cycle decompositions are: σ = (135)(24) τ = (15)(23)(4) σ2 = (153)(2)(4) στ = (1)(2534) τσ = (1243)(5) τ 2σ = (135)(24): 1.3.7* Write out the cycle decomposition of each element of order 2 in S4. Elements of order 2 are also called involutions. There are six formed from a single transposition, (12); (13); (14); (23); (24); (34), and three from pairs of transpositions: (12)(34); (13)(24); (14)(23). 3.1.6* Define ' : R× ! {±1g by letting '(x) be x divided by the absolute value of x. Describe the fibers of ' and prove that ' is a homomorphism. The fibers of ' are '−1(1) = (0; 1) = fall positive realsg and '−1(−1) = (−∞; 0) = fall negative realsg. 3.1.7* Define π : R2 ! R by π((x; y)) = x + y. Prove that π is a surjective homomorphism and describe the kernel and fibers of π geometrically. The map π is surjective because e.g. π((x; 0)) = x. The kernel of π is the line y = −x through the origin. -
An Introduction to Operad Theory
AN INTRODUCTION TO OPERAD THEORY SAIMA SAMCHUCK-SCHNARCH Abstract. We give an introduction to category theory and operad theory aimed at the undergraduate level. We first explore operads in the category of sets, and then generalize to other familiar categories. Finally, we develop tools to construct operads via generators and relations, and provide several examples of operads in various categories. Throughout, we highlight the ways in which operads can be seen to encode the properties of algebraic structures across different categories. Contents 1. Introduction1 2. Preliminary Definitions2 2.1. Algebraic Structures2 2.2. Category Theory4 3. Operads in the Category of Sets 12 3.1. Basic Definitions 13 3.2. Tree Diagram Visualizations 14 3.3. Morphisms and Algebras over Operads of Sets 17 4. General Operads 22 4.1. Basic Definitions 22 4.2. Morphisms and Algebras over General Operads 27 5. Operads via Generators and Relations 33 5.1. Quotient Operads and Free Operads 33 5.2. More Examples of Operads 38 5.3. Coloured Operads 43 References 44 1. Introduction Sets equipped with operations are ubiquitous in mathematics, and many familiar operati- ons share key properties. For instance, the addition of real numbers, composition of functions, and concatenation of strings are all associative operations with an identity element. In other words, all three are examples of monoids. Rather than working with particular examples of sets and operations directly, it is often more convenient to abstract out their common pro- perties and work with algebraic structures instead. For instance, one can prove that in any monoid, arbitrarily long products x1x2 ··· xn have an unambiguous value, and thus brackets 2010 Mathematics Subject Classification. -
Categories, Functors, and Natural Transformations I∗
Lecture 2: Categories, functors, and natural transformations I∗ Nilay Kumar June 4, 2014 (Meta)categories We begin, for the moment, with rather loose definitions, free from the technicalities of set theory. Definition 1. A metagraph consists of objects a; b; c; : : :, arrows f; g; h; : : :, and two operations, as follows. The first is the domain, which assigns to each arrow f an object a = dom f, and the second is the codomain, which assigns to each arrow f an object b = cod f. This is visually indicated by f : a ! b. Definition 2. A metacategory is a metagraph with two additional operations. The first is the identity, which assigns to each object a an arrow Ida = 1a : a ! a. The second is the composition, which assigns to each pair g; f of arrows with dom g = cod f an arrow g ◦ f called their composition, with g ◦ f : dom f ! cod g. This operation may be pictured as b f g a c g◦f We require further that: composition is associative, k ◦ (g ◦ f) = (k ◦ g) ◦ f; (whenever this composition makese sense) or diagrammatically that the diagram k◦(g◦f)=(k◦g)◦f a d k◦g f k g◦f b g c commutes, and that for all arrows f : a ! b and g : b ! c, we have 1b ◦ f = f and g ◦ 1b = g; or diagrammatically that the diagram f a b f g 1b g b c commutes. ∗This talk follows [1] I.1-4 very closely. 1 Recall that a diagram is commutative when, for each pair of vertices c and c0, any two paths formed from direct edges leading from c to c0 yield, by composition of labels, equal arrows from c to c0.