Equivalence Relations and Partitions, Normal Subgroups, Quotient Groups, and Homomorphisms


 Oliver Ferguson
 3 years ago
 Views:
Transcription
1 Equivalence Relations and Partitions, Normal Subgroups, Quotient Groups, and Homomorphisms Math 356 Abstract We sum up the main features of our last three class sessions, which list of topics are given above. Interspersed throughout these notes are exercises, which are due Tuesday, March 6. Any problem with an (H) behind it is required homework, anything with an (S) is suggested for yourself, and should not be handed in (although I will be happy to go over these problems with you in office hours). There are quite a few problems here, but the six problems that are to be handed in are mostly very short. 1
2 1 Equivalence relations and partitions Throughout this section, let A and B represent fixed sets. Whenever we refer to a symbol of the form a or a, then we are refering to an element of A (even if there is no subscript), and any symbol of the form b, or b will refer to an element of B. (In this notation, we will replace the by any index value or variable.) We will also assume that the symbols for subset, union, and intersection (,, and respectively) are understood. We can define a new set from A and B, the cartesian product or direct product of A and B, denoted A B, as follows. A B = {(a, b) a A, B B} The set A B is the set of all ordered pairs, whose first coordinates come from the set A, and whose second coordinates come from the set B. We define a relation R between A and B to be a subset of A B, that is R A B. The idea here is that we think that a is related to b by R if and only if (a, b) R. For example, a function f from A to B is a special form of relation that satisfies two further rules. 1. For all a A, there is b B so that (a, b) f. 2. For all a A, if (a, b 1 ) f and (a, b 2 ) f, then b 1 = b 2. Note that in this point of view, a function IS what we usually think of as a graph of a function! Sometimes we focus on relations from a set to itself. For example, we can define the relation < on R by the rule <= {(r 1, r 2 ) r 2 r 1 = r2 r 1 andr 2 r 1 }. There is actually a cheat here, since it is very hard to define the absolute value of something, if you cannot tell if it is greater than zero! Another sort of relation is an equivalence relation. An equivalence relation on A is a subset E of A A which satisfies three rules. 1. For all a A we have that (a, a) E (Reflexivity Rule). 2. If (a 1, a 2 ) E then (a 2, a 1 ) E (Symmetry Rule). 3. If (a 1, a 2 ), (a 2, a 3 ) E then (a 1, a 3 ) E (Transitivity Rule). Exercise 1 (H) We will say that two points in the plane are related if they are on the graph of a line of the form y = 2x + c for some constant c. Does this define an equivalence relation on the plane? What if we use the rule that two points are related if they are both in a line of the form y = cx for some constant c? We often denote the set of an equivalence relation with the symbol, and we denote the fact that (a 1, a 2 ) by the sequence a 1 a 2. Suppose that is an equivalence relation on A, and a A, then we can build the set a = {a A a a }, which is the set of all things in a which are related (equivalent) to a under. This is often called the equivalence class of a. Note that the notation a is a little nonstandard, but the use of the preposition under seems quite common. 1
3 Exercise 2 (H) Suppose is an equivalence relation on A, and that a, b A. If a b, then a = b The set of sets, whose elements are the distinct equivalence classes of the elements of A under, is a partition of A. That is, a collection of disjoint subsets of A whose union is all of A. It will only take a moment s thought to see that given a partition of A, one can define an equivalence relation on A whose equivalence classes are the elements of the partition (note that an element of a partition of A is a subset of A). Exercise 3 (H) For each relation in the first exercise that is an equivalence relation, describe the equivalence classes. 2
4 2 Normal subgroups and quotient groups For this section, we will refer to a fixed group G, and its subgroup N. We will adopt the convention that symbols of the form g or g are elements of G, and n and n are elements of N. The symbol e will always denote the identity of G (and therefore of N as well). Given two subsets R and S of G, we can define a third subset RS using the binary operation of G as follows. RS = {rs r R, s S} This enables us to define a Multiplication on subsets of G. If R = {r}, then we will often denote the set RS = {rs s S} by the symbol rs instead. Likewise if S = {s}, then we may denote RS by Rs = {rs r R}. As multiplication in G is associative, we will assume the notation g 1 Ng 2 is now understood, given elements g 1, g 2 G. If H is a subgroup of G, and g is an element of G, then we call the set gh a left coset of H in G. We say that N is a normal subgroup of G if and only if, given any element g G, we have gng 1 = N. We will denote that N is a normal subgroup of G by the notation N G. Exercise 4 (H) Suppose g G, and n 1 N, where N G. Show that there is n 2 N so that gn 1 = n 2 g. Exercise 5 (H) Suppose N G, and g 1, g 2 G. Show that if g 1 N g 2 N, then g 1 g 2 N. Conclude that either g 1 N g 2 N =, or g 1 N = g 2 N. Consider the set Q = {gn g G} of left cosets of N in G. Recalling that sets do not allow duplicates, we see that Q usually has fewer elements than G. From the previous exercise, we see that Q is a partition of G. Exercise 6 (H) Show that the sets g 1 Ng 2 N and g 1 g 2 N are equal. Conclude that Q, under the binary operation g 1 Ng 2 N = (g 1 g 2 )N forms a group. (You will need to discuss closure, the existence of an identity, inverses, and associativity.) Q is called the Quotient group of G by N, or a quotient group of G, for short. It is often denoted by the notation Q = G/N, which is read as Q is G mod N or Q is the quotient of G by N, or various similar language. Exercise 7 (S) Is there a quotient group of order 3 from S 3? In other words, can you find a normal subgroup of size two in S 3? Exercise 8 (S) Give an example of the fact that if H G, where H is not normal in G, then the left cosets of H in G do not form a group under coset multiplication. 3
5 3 Homomorphisms We will delve much deeper into this subject as the course goes on. For now, I simply give a short explanation, definition, and a collection of exercises/facts from class. Given any class of mathematical objects, such as sets, vector spaces, or topological spaces for example, we like to be able to relate these objects to each other. For example, functions that map sets into each other, linear transformations that map vector spaces into each other, and continuous maps that map topological spaces into each other. These special types of mapping relations have one chief aspect in common; each of these operations works as a set function on the underlying sets, but they also preserve the extra structure of the object to some extent. This is not saying much in the world of sets, but for example, linear transformations preserve linear combinations, that is, they do not send lines to curvy curves! Continuous functions do not rip apart points that are close together. (Although, continuous functions may stretch a region between nearby points over a large range, in order to have points which looked close together map to places far apart. This is just a reflection of the idea that the points we started with were not Close enough together to stay together.) When the objects we are concerned with are groups, the map we use is called a homomorphism. A homomorphism between two groups G and H is a set map between the underlying sets that respects the group product operations of the two groups. More formally, we have the the following. Let (G, ) and (H, ) be groups. A function φ : G H is a homomorphism if for all g 1, g 2 G, we have that φ(g 1 g 2 ) = φ(g 1 ) φ(g 2 ). We will typically drop the explicit use of the two group operations from our notation in the future, I only include it in this definition explicitly to emphasize that the two different products are occurring in the two different groups. The homomorphism phi can be thought of as interpreting a multiplication in the group G as a multiplication in the group H. This last approach is often useful if H is simpler than G to understand; we throw away some extra information, hopefully preserving enough of the structure of G within H to answer questions about G! Where there is a homomorphism φ : G H, there is an important subset of G, called the kernel of φ, which is often denoted Ker(φ). It is defined as follows. Ker(φ) = {g G φg = e H } Here I am using e H to denote the identity element of H. The kernel of a homomorphism is very similar, in concept, to the kernel of a linear transformation from linear algebra, and just as important. Here is a list of facts about homomorphisms from class. Any fact with an (S) at the beginning was not shown in class, however, it would be good if you try to prove these facts on your own. Remark 3.1 Let φ : G H be a group homomorphism. 1. φ(g n ) = φ(g) n for any integer n (here a negative integer means the inverse taken to a positive power). 4
6 2. Ker(φ) is a subgroup of G. 3. (S) Ker(φ) is a normal subgroup of G. 4. (S) If R G, then φ(r) = {h H r R, φ(r) = h} is a subgroup of H. 5. (S) If S H, then φ 1 (S) = {g G φ(g) S}, is a subgroup of G. Here is one more suggested problem for fun... Let K4 = a, b a 2, b 2, a 1 b 1 ab be a group, (this group is called the Kleinfour group, recall that the relation [a, b] = a 1 b 1 ab means that a and b commute). Can you find a homomorphism from D 6 onto K4? (Hint: the kernel of the homomorphism must be a normal subgroup of D 6 with three elements.) 5
Kevin James. Quotient Groups and Homomorphisms: Definitions and Examp
Quotient Groups and Homomorphisms: Definitions and Examples Definition If φ : G H is a homomorphism of groups, the kernel of φ is the set ker(φ){g G φ(g) = 1 H }. Definition If φ : G H is a homomorphism
More informationDefinition List Modern Algebra, Fall 2011 Anders O.F. Hendrickson
Definition List Modern Algebra, Fall 2011 Anders O.F. Hendrickson On almost every Friday of the semester, we will have a brief quiz to make sure you have memorized the definitions encountered in our studies.
More informationDefinitions. Notations. Injective, Surjective and Bijective. Divides. Cartesian Product. Relations. Equivalence Relations
Page 1 Definitions Tuesday, May 8, 2018 12:23 AM Notations " " means "equals, by definition" the set of all real numbers the set of integers Denote a function from a set to a set by Denote the image of
More informationLecture 4: Constructing the Integers, Rationals and Reals
Math/CS 20: Intro. to Math Professor: Padraic Bartlett Lecture 4: Constructing the Integers, Rationals and Reals Week 5 UCSB 204 The Integers Normally, using the natural numbers, you can easily define
More informationLecture 7.3: Ring homomorphisms
Lecture 7.3: Ring homomorphisms Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4120, Modern Algebra M. Macauley (Clemson) Lecture 7.3:
More information1 2 3 style total. Circle the correct answer; no explanation is required. Each problem in this section counts 5 points.
1 2 3 style total Math 415 Examination 3 Please print your name: Answer Key 1 True/false Circle the correct answer; no explanation is required. Each problem in this section counts 5 points. 1. The rings
More informationSection 15 Factorgroup computation and simple groups
Section 15 Factorgroup computation and simple groups Instructor: Yifan Yang Fall 2006 Outline Factorgroup computation Simple groups The problem Problem Given a factor group G/H, find an isomorphic group
More informationMATH RING ISOMORPHISM THEOREMS
MATH 371  RING ISOMORPHISM THEOREMS DR. ZACHARY SCHERR 1. Theory In this note we prove all four isomorphism theorems for rings, and provide several examples on how they get used to describe quotient rings.
More informationAN ALGEBRA PRIMER WITH A VIEW TOWARD CURVES OVER FINITE FIELDS
AN ALGEBRA PRIMER WITH A VIEW TOWARD CURVES OVER FINITE FIELDS The integers are the set 1. Groups, Rings, and Fields: Basic Examples Z := {..., 3, 2, 1, 0, 1, 2, 3,...}, and we can add, subtract, and multiply
More information6 Cosets & Factor Groups
6 Cosets & Factor Groups The course becomes markedly more abstract at this point. Our primary goal is to break apart a group into subsets such that the set of subsets inherits a natural group structure.
More informationSymmetries and Polynomials
Symmetries and Polynomials Aaron Landesman and Apurva Nakade June 30, 2018 Introduction In this class we ll learn how to solve a cubic. We ll also sketch how to solve a quartic. We ll explore the connections
More informationProblem 1. Let I and J be ideals in a ring commutative ring R with 1 R. Recall
I. TakeHome Portion: Math 350 Final Exam Due by 5:00pm on Tues. 5/12/15 No resources/devices other than our class textbook and class notes/handouts may be used. You must work alone. Choose any 5 problems
More informationModern Algebra Prof. Manindra Agrawal Department of Computer Science and Engineering Indian Institute of Technology, Kanpur
Modern Algebra Prof. Manindra Agrawal Department of Computer Science and Engineering Indian Institute of Technology, Kanpur Lecture 02 Groups: Subgroups and homomorphism (Refer Slide Time: 00:13) We looked
More informationREU 2007 Discrete Math Lecture 2
REU 2007 Discrete Math Lecture 2 Instructor: László Babai Scribe: Shawn Drenning June 19, 2007. Proofread by instructor. Last updated June 20, 1 a.m. Exercise 2.0.1. Let G be an abelian group and A G be
More informationMath 345 Sp 07 Day 7. b. Prove that the image of a homomorphism is a subring.
Math 345 Sp 07 Day 7 1. Last time we proved: a. Prove that the kernel of a homomorphism is a subring. b. Prove that the image of a homomorphism is a subring. c. Let R and S be rings. Suppose R and S are
More informationDEPARTMENT OF MATHEMATIC EDUCATION MATHEMATIC AND NATURAL SCIENCE FACULTY
HANDOUT ABSTRACT ALGEBRA MUSTHOFA DEPARTMENT OF MATHEMATIC EDUCATION MATHEMATIC AND NATURAL SCIENCE FACULTY 2012 BINARY OPERATION We are all familiar with addition and multiplication of two numbers. Both
More information2. normal subgroup and quotient group We begin by stating a couple of elementary lemmas Lemma. Let A and B be sets and f : A B be an onto
2. normal subgroup and quotient group We begin by stating a couple of elementary lemmas. 2.1. Lemma. Let A and B be sets and f : A B be an onto function. For b B, recall that f 1 (b) ={a A: f(a) =b}. LetF
More informationExercises on chapter 0
Exercises on chapter 0 1. A partially ordered set (poset) is a set X together with a relation such that (a) x x for all x X; (b) x y and y x implies that x = y for all x, y X; (c) x y and y z implies that
More informationMA441: Algebraic Structures I. Lecture 26
MA441: Algebraic Structures I Lecture 26 10 December 2003 1 (page 179) Example 13: A 4 has no subgroup of order 6. BWOC, suppose H < A 4 has order 6. Then H A 4, since it has index 2. Thus A 4 /H has order
More informationChapter 3. Introducing Groups
Chapter 3 Introducing Groups We need a supermathematics in which the operations are as unknown as the quantities they operate on, and a supermathematician who does not know what he is doing when he performs
More information(Think: three copies of C) i j = k = j i, j k = i = k j, k i = j = i k.
Warmup: The quaternion group, denoted Q 8, is the set {1, 1, i, i, j, j, k, k} with product given by 1 a = a 1 = a a Q 8, ( 1) ( 1) = 1, i 2 = j 2 = k 2 = 1, ( 1) a = a ( 1) = a a Q 8, (Think: three copies
More informationABSTRACT ALGEBRA 1, LECTURE NOTES 5: HOMOMORPHISMS, ISOMORPHISMS, SUBGROUPS, QUOTIENT ( FACTOR ) GROUPS. ANDREW SALCH
ABSTRACT ALGEBRA 1, LECTURE NOTES 5: HOMOMORPHISMS, ISOMORPHISMS, SUBGROUPS, QUOTIENT ( FACTOR ) GROUPS. ANDREW SALCH 1. Homomorphisms and isomorphisms between groups. Definition 1.1. Let G, H be groups.
More informationAlgebraic Structures Exam File Fall 2013 Exam #1
Algebraic Structures Exam File Fall 2013 Exam #1 1.) Find all four solutions to the equation x 4 + 16 = 0. Give your answers as complex numbers in standard form, a + bi. 2.) Do the following. a.) Write
More informationModern Algebra I. Circle the correct answer; no explanation is required. Each problem in this section counts 5 points.
1 2 3 style total Math 415 Please print your name: Answer Key 1 True/false Circle the correct answer; no explanation is required. Each problem in this section counts 5 points. 1. Every group of order 6
More informationRings of Residues. S. F. Ellermeyer. September 18, ; [1] m
Rings of Residues S F Ellermeyer September 18, 2006 If m is a positive integer, then we obtain the partition C = f[0] m ; [1] m ; : : : ; [m 1] m g of Z into m congruence classes (This is discussed in
More informationBasic Definitions: Group, subgroup, order of a group, order of an element, Abelian, center, centralizer, identity, inverse, closed.
Math 546 Review Exam 2 NOTE: An (*) at the end of a line indicates that you will not be asked for the proof of that specific item on the exam But you should still understand the idea and be able to apply
More information1. To be a grandfather. Objects of our consideration are people; a person a is associated with a person b if a is a grandfather of b.
20 [1610161020 ] 3.3 Binary relations In mathematics, as in everyday situations, we often speak about a relationship between objects, which means an idea of two objects being related or associated one
More informationNotes on Sets, Relations and Functions
PURE MTH 3002 Topology & Analysis III (3246) 2002 Notes on Sets, Relations and Functions These are some notes taken from Mathematical Applications (now Mathematics for Information Technology (MIT)). They
More information2) e = e G G such that if a G 0 =0 G G such that if a G e a = a e = a. 0 +a = a+0 = a.
Chapter 2 Groups Groups are the central objects of algebra. In later chapters we will define rings and modules and see that they are special cases of groups. Also ring homomorphisms and module homomorphisms
More informationModern Algebra Prof. Manindra Agrawal Department of Computer Science and Engineering Indian Institute of Technology, Kanpur
Modern Algebra Prof. Manindra Agrawal Department of Computer Science and Engineering Indian Institute of Technology, Kanpur Lecture  05 Groups: Structure Theorem So, today we continue our discussion forward.
More information2MA105 Algebraic Structures I
2MA105 Algebraic Structures I PerAnders Svensson http://homepage.lnu.se/staff/psvmsi/2ma105.html Lecture 7 Cosets once again Factor Groups Some Properties of Factor Groups Homomorphisms November 28, 2011
More informationCosets and Normal Subgroups
Cosets and Normal Subgroups (Last Updated: November 3, 2017) These notes are derived primarily from Abstract Algebra, Theory and Applications by Thomas Judson (16ed). Most of this material is drawn from
More informationDiscrete Mathematics. Benny George K. September 22, 2011
Discrete Mathematics Benny George K Department of Computer Science and Engineering Indian Institute of Technology Guwahati ben@iitg.ernet.in September 22, 2011 Set Theory Elementary Concepts Let A and
More informationExtra exercises for algebra
Extra exercises for algebra These are extra exercises for the course algebra. They are meant for those students who tend to have already solved all the exercises at the beginning of the exercise session
More informationMATH 101: ALGEBRA I WORKSHEET, DAY #1. We review the prerequisites for the course in set theory and beginning a first pass on group. 1.
MATH 101: ALGEBRA I WORKSHEET, DAY #1 We review the prerequisites for the course in set theory and beginning a first pass on group theory. Fill in the blanks as we go along. 1. Sets A set is a collection
More informationLecture 4.1: Homomorphisms and isomorphisms
Lecture 4.: Homomorphisms and isomorphisms Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4, Modern Algebra M. Macauley (Clemson) Lecture
More informationLecture 3. Theorem 1: D 6
Lecture 3 This week we have a longer section on homomorphisms and isomorphisms and start formally working with subgroups even though we have been using them in Chapter 1. First, let s finish what was claimed
More informationINTRODUCTION TO THE GROUP THEORY
Lecture Notes on Structure of Algebra INTRODUCTION TO THE GROUP THEORY By : Drs. Antonius Cahya Prihandoko, M.App.Sc email: antoniuscp.fkip@unej.ac.id Mathematics Education Study Program Faculty of Teacher
More informationEXERCISES ON THE OUTER AUTOMORPHISMS OF S 6
EXERCISES ON THE OUTER AUTOMORPHISMS OF S 6 AARON LANDESMAN 1. INTRODUCTION In this class, we investigate the outer automorphism of S 6. Let s recall some definitions, so that we can state what an outer
More informationCategory Theory. Categories. Definition.
Category Theory Category theory is a general mathematical theory of structures, systems of structures and relationships between systems of structures. It provides a unifying and economic mathematical modeling
More informationGroup Theory, Lattice Geometry, and Minkowski s Theorem
Group Theory, Lattice Geometry, and Minkowski s Theorem Jason Payne Physics and mathematics have always been inextricably interwoven one s development and progress often hinges upon the other s. It is
More informationGROUPS AND THEIR REPRESENTATIONS. 1. introduction
GROUPS AND THEIR REPRESENTATIONS KAREN E. SMITH 1. introduction Representation theory is the study of the concrete ways in which abstract groups can be realized as groups of rigid transformations of R
More informationCONSEQUENCES OF THE SYLOW THEOREMS
CONSEQUENCES OF THE SYLOW THEOREMS KEITH CONRAD For a group theorist, Sylow s Theorem is such a basic tool, and so fundamental, that it is used almost without thinking, like breathing. Geoff Robinson 1.
More informationEquivalence Relations
Equivalence Relations Definition 1. Let X be a nonempty set. A subset E X X is called an equivalence relation on X if it satisfies the following three properties: 1. Reflexive: For all x X, (x, x) E.
More informationFINAL EXAM MATH 150A FALL 2016
FINAL EXAM MATH 150A FALL 2016 The final exam consists of eight questions, each worth 10 or 15 points. The maximum score is 100. You are not allowed to use books, calculators, mobile phones or anything
More informationError Correcting Codes Prof. Dr. P Vijay Kumar Department of Electrical Communication Engineering Indian Institute of Science, Bangalore
(Refer Slide Time: 00:54) Error Correcting Codes Prof. Dr. P Vijay Kumar Department of Electrical Communication Engineering Indian Institute of Science, Bangalore Lecture No. # 05 Cosets, Rings & Fields
More informationMATH 28A MIDTERM 2 INSTRUCTOR: HAROLD SULTAN
NAME: MATH 28A MIDTERM 2 INSTRUCTOR: HAROLD SULTAN 1. INSTRUCTIONS (1) Timing: You have 80 minutes for this midterm. (2) Partial Credit will be awarded. Please show your work and provide full solutions,
More informationABSTRACT ALGEBRA 1, LECTURES NOTES 5: SUBGROUPS, CONJUGACY, NORMALITY, QUOTIENT GROUPS, AND EXTENSIONS.
ABSTRACT ALGEBRA 1, LECTURES NOTES 5: SUBGROUPS, CONJUGACY, NORMALITY, QUOTIENT GROUPS, AND EXTENSIONS. ANDREW SALCH 1. Subgroups, conjugacy, normality. I think you already know what a subgroup is: Definition
More informationLectures  XXIII and XXIV Coproducts and Pushouts
Lectures  XXIII and XXIV Coproducts and Pushouts We now discuss further categorical constructions that are essential for the formulation of the Seifert Van Kampen theorem. We first discuss the notion
More informationBASIC GROUP THEORY : G G G,
BASIC GROUP THEORY 18.904 1. Definitions Definition 1.1. A group (G, ) is a set G with a binary operation : G G G, and a unit e G, possessing the following properties. (1) Unital: for g G, we have g e
More informationScott Taylor 1. EQUIVALENCE RELATIONS. Definition 1.1. Let A be a set. An equivalence relation on A is a relation such that:
Equivalence MA Relations 274 and Partitions Scott Taylor 1. EQUIVALENCE RELATIONS Definition 1.1. Let A be a set. An equivalence relation on A is a relation such that: (1) is reflexive. That is, (2) is
More informationA Little Beyond: Linear Algebra
A Little Beyond: Linear Algebra Akshay Tiwary March 6, 2016 Any suggestions, questions and remarks are welcome! 1 A little extra Linear Algebra 1. Show that any set of nonzero polynomials in [x], no two
More informationPart IV. Rings and Fields
IV.18 Rings and Fields 1 Part IV. Rings and Fields Section IV.18. Rings and Fields Note. Roughly put, modern algebra deals with three types of structures: groups, rings, and fields. In this section we
More informationCS 468: Computational Topology Group Theory Fall b c b a b a c b a c b c c b a
Q: What s purple and commutes? A: An abelian grape! Anonymous Group Theory Last lecture, we learned about a combinatorial method for characterizing spaces: using simplicial complexes as triangulations
More informationCosets, factor groups, direct products, homomorphisms, isomorphisms
Cosets, factor groups, direct products, homomorphisms, isomorphisms Sergei Silvestrov Spring term 2011, Lecture 11 Contents of the lecture Cosets and the theorem of Lagrange. Direct products and finitely
More informationMATH 101: ALGEBRA I WORKSHEET, DAY #3. Fill in the blanks as we finish our first pass on prerequisites of group theory.
MATH 101: ALGEBRA I WORKSHEET, DAY #3 Fill in the blanks as we finish our first pass on prerequisites of group theory 1 Subgroups, cosets Let G be a group Recall that a subgroup H G is a subset that is
More informationCOMBINATORIAL GROUP THEORY NOTES
COMBINATORIAL GROUP THEORY NOTES These are being written as a companion to Chapter 1 of Hatcher. The aim is to give a description of some of the group theory required to work with the fundamental groups
More informationSupplement. Dr. Bob s Modern Algebra Glossary Based on Fraleigh s A First Course on Abstract Algebra, 7th Edition, Sections 0 through IV.
Glossary 1 Supplement. Dr. Bob s Modern Algebra Glossary Based on Fraleigh s A First Course on Abstract Algebra, 7th Edition, Sections 0 through IV.23 Abelian Group. A group G, (or just G for short) is
More informationMath Studies Algebra II
Math Studies Algebra II Prof. Clinton Conley Spring 2017 Contents 1 January 18, 2017 4 1.1 Logistics..................................................... 4 1.2 Modules.....................................................
More informationYour Name MATH 435, EXAM #1
MATH 435, EXAM #1 Your Name You have 50 minutes to do this exam. No calculators! No notes! For proofs/justifications, please use complete sentences and make sure to explain any steps which are questionable.
More informationMATH ABSTRACT ALGEBRA DISCUSSIONS  WEEK 8
MAT 410  ABSTRACT ALEBRA DISCUSSIONS  WEEK 8 CAN OZAN OUZ 1. Isomorphism Theorems In group theory, there are three main isomorphism theorems. They all follow from the first isomorphism theorem. Let s
More information2a 2 4ac), provided there is an element r in our
MTH 310002 Test II Review Spring 2012 Absractions versus examples The purpose of abstraction is to reduce ideas to their essentials, uncluttered by the details of a specific situation Our lectures built
More informationUMASS AMHERST MATH 300 SP 05, F. HAJIR HOMEWORK 8: (EQUIVALENCE) RELATIONS AND PARTITIONS
UMASS AMHERST MATH 300 SP 05, F. HAJIR HOMEWORK 8: (EQUIVALENCE) RELATIONS AND PARTITIONS 1. Relations Recall the concept of a function f from a source set X to a target set Y. It is a rule for mapping
More informationLecture Note of Week 2
Lecture Note of Week 2 2. Homomorphisms and Subgroups (2.1) Let G and H be groups. A map f : G H is a homomorphism if for all x, y G, f(xy) = f(x)f(y). f is an isomorphism if it is bijective. If f : G
More informationUnit 1: Equations & Inequalities in One Variable
Date Period Unit 1: Equations & Inequalities in One Variable Day Topic 1 Properties of Real Numbers Algebraic Expressions Solving Equations 3 Solving Inequalities 4 QUIZ 5 Absolute Value Equations 6 Double
More informationISOMORPHISMS KEITH CONRAD
ISOMORPHISMS KEITH CONRAD 1. Introduction Groups that are not literally the same may be structurally the same. An example of this idea from high school math is the relation between multiplication and addition
More informationMATH 430 PART 2: GROUPS AND SUBGROUPS
MATH 430 PART 2: GROUPS AND SUBGROUPS Last class, we encountered the structure D 3 where the set was motions which preserve an equilateral triangle and the operation was function composition. We determined
More informationNormal Subgroups and Quotient Groups
Normal Subgroups and Quotient Groups 3202014 A subgroup H < G is normal if ghg 1 H for all g G. Notation: H G. Every subgroup of an abelian group is normal. Every subgroup of index 2 is normal. If H
More informationRelations and Equivalence Relations
Relations and Equivalence Relations In this section, we shall introduce a formal definition for the notion of a relation on a set. This is something we often take for granted in elementary algebra courses,
More informationNOTES FOR DRAGOS: MATH 210 CLASS 12, THURS. FEB. 22
NOTES FOR DRAGOS: MATH 210 CLASS 12, THURS. FEB. 22 RAVI VAKIL Hi Dragos The class is in 381T, 1:15 2:30. This is the very end of Galois theory; you ll also start commutative ring theory. Tell them: midterm
More information3.8 Cosets, Normal Subgroups, and Factor Groups
3.8 J.A.Beachy 1 3.8 Cosets, Normal Subgroups, and Factor Groups from A Study Guide for Beginner s by J.A.Beachy, a supplement to Abstract Algebra by Beachy / Blair 29. Define φ : C R by φ(z) = z, for
More information2. The center of G, denoted by Z(G), is the abelian subgroup which commutes with every elements of G. The center always contains the unit element e.
Chapter 2 Group Structure To be able to use groups in physics, or mathematics, we need to know what are the important features distinguishing one group from another. This is under the heading of group
More informationSUPPLEMENT ON THE SYMMETRIC GROUP
SUPPLEMENT ON THE SYMMETRIC GROUP RUSS WOODROOFE I presented a couple of aspects of the theory of the symmetric group S n differently than what is in Herstein. These notes will sketch this material. You
More information3. G. Groups, as men, will be known by their actions.  Guillermo Moreno
3.1. The denition. 3. G Groups, as men, will be known by their actions.  Guillermo Moreno D 3.1. An action of a group G on a set X is a function from : G X! X such that the following hold for all g, h
More informationUMASS AMHERST MATH 411 SECTION 2, FALL 2009, F. HAJIR
UMASS AMHERST MATH 411 SECTION 2, FALL 2009, F. HAJIR HOMEWORK 2: DUE TH. OCT. 1 READINGS: These notes are intended as a supplement to the textbook, not a replacement for it. 1. Elements of a group, their
More informationFall 2014 Math 122 Midterm 1
1. Some things you ve (maybe) done before. 5 points each. (a) If g and h are elements of a group G, show that (gh) 1 = h 1 g 1. (gh)(h 1 g 1 )=g(hh 1 )g 1 = g1g 1 = gg 1 =1. Likewise, (h 1 g 1 )(gh) =h
More informationENTRY GROUP THEORY. [ENTRY GROUP THEORY] Authors: started Mark Lezama: October 2003 Literature: Algebra by Michael Artin, Mathworld.
ENTRY GROUP THEORY [ENTRY GROUP THEORY] Authors: started Mark Lezama: October 2003 Literature: Algebra by Michael Artin, Mathworld Group theory [Group theory] is studies algebraic objects called groups.
More information2. Introduction to commutative rings (continued)
2. Introduction to commutative rings (continued) 2.1. New examples of commutative rings. Recall that in the first lecture we defined the notions of commutative rings and field and gave some examples of
More informationYale University Department of Mathematics Math 350 Introduction to Abstract Algebra Fall Midterm Exam Review Solutions
Yale University Department of Mathematics Math 350 Introduction to Abstract Algebra Fall 2015 Midterm Exam Review Solutions Practice exam questions: 1. Let V 1 R 2 be the subset of all vectors whose slope
More informationMath 210B:Algebra, Homework 2
Math 210B:Algebra, Homework 2 Ian Coley January 21, 2014 Problem 1. Is f = 2X 5 6X + 6 irreducible in Z[X], (S 1 Z)[X], for S = {2 n, n 0}, Q[X], R[X], C[X]? To begin, note that 2 divides all coefficients
More informationHOMEWORK 3 LOUISPHILIPPE THIBAULT
HOMEWORK 3 LOUISPHILIPPE THIBAULT Problem 1 Let G be a group of order 56. We have that 56 = 2 3 7. Then, using Sylow s theorem, we have that the only possibilities for the number of Sylowp subgroups
More informationExploring the Exotic Setting for Algebraic Geometry
Exploring the Exotic Setting for Algebraic Geometry Victor I. Piercey University of Arizona Integration Workshop Project August 610, 2010 1 Introduction In this project, we will describe the basic topology
More informationPermutation groups/1. 1 Automorphism groups, permutation groups, abstract
Permutation groups Whatever you have to do with a structureendowed entity Σ try to determine its group of automorphisms... You can expect to gain a deep insight into the constitution of Σ in this way.
More informationHomework 10 M 373K by Mark Lindberg (mal4549)
Homework 10 M 373K by Mark Lindberg (mal4549) 1. Artin, Chapter 11, Exercise 1.1. Prove that 7 + 3 2 and 3 + 5 are algebraic numbers. To do this, we must provide a polynomial with integer coefficients
More informationGROUP ACTIONS RYAN C. SPIELER
GROUP ACTIONS RYAN C. SPIELER Abstract. In this paper, we examine group actions. Groups, the simplest objects in Algebra, are sets with a single operation. We will begin by defining them more carefully
More informationINVERSE LIMITS AND PROFINITE GROUPS
INVERSE LIMITS AND PROFINITE GROUPS BRIAN OSSERMAN We discuss the inverse limit construction, and consider the special case of inverse limits of finite groups, which should best be considered as topological
More informationLecture 6 Special Subgroups
Lecture 6 Special Subgroups Review: Recall, for any homomorphism ϕ : G H, the kernel of ϕ is and the image of ϕ is ker(ϕ) = {g G ϕ(g) = e H } G img(ϕ) = {h H ϕ(g) = h for some g G} H. (They are subgroups
More information1 Fields and vector spaces
1 Fields and vector spaces In this section we revise some algebraic preliminaries and establish notation. 1.1 Division rings and fields A division ring, or skew field, is a structure F with two binary
More informationGroups and Symmetries
Groups and Symmetries Definition: Symmetry A symmetry of a shape is a rigid motion that takes vertices to vertices, edges to edges. Note: A rigid motion preserves angles and distances. Definition: Group
More informationRings. Chapter 1. Definition 1.2. A commutative ring R is a ring in which multiplication is commutative. That is, ab = ba for all a, b R.
Chapter 1 Rings We have spent the term studying groups. A group is a set with a binary operation that satisfies certain properties. But many algebraic structures such as R, Z, and Z n come with two binary
More informationRecall: Properties of Homomorphisms
Recall: Properties of Homomorphisms Let φ : G Ḡ be a homomorphism, let g G, and let H G. Properties of elements Properties of subgroups 1. φ(e G ) = eḡ 1. φ(h) Ḡ. 2. φ(g n ) = (φ(g)) n for all n Z. 2.
More informationMath 222A W03 D. Congruence relations
Math 222A W03 D. 1. The concept Congruence relations Let s start with a familiar case: congruence mod n on the ring Z of integers. Just to be specific, let s use n = 6. This congruence is an equivalence
More informationMATH HL OPTION  REVISION SETS, RELATIONS AND GROUPS Compiled by: Christos Nikolaidis
MATH HL OPTION  REVISION SETS, RELATIONS AND GROUPS Compiled by: Christos Nikolaidis PART B: GROUPS GROUPS 1. ab The binary operation a * b is defined by a * b = a+ b +. (a) Prove that * is associative.
More informationEquational Logic. Chapter Syntax Terms and Term Algebras
Chapter 2 Equational Logic 2.1 Syntax 2.1.1 Terms and Term Algebras The natural logic of algebra is equational logic, whose propositions are universally quantified identities between terms built up from
More informationEXAMPLES AND EXERCISES IN BASIC CATEGORY THEORY
EXAMPLES AND EXERCISES IN BASIC CATEGORY THEORY 1. Categories 1.1. Generalities. I ve tried to be as consistent as possible. In particular, throughout the text below, categories will be denoted by capital
More informationEXAM 3 MAT 423 Modern Algebra I Fall c d a + c (b + d) d c ad + bc ac bd
EXAM 3 MAT 23 Modern Algebra I Fall 201 Name: Section: I All answers must include either supporting work or an explanation of your reasoning. MPORTANT: These elements are considered main part of the answer
More informationMA554 Assessment 1 Cosets and Lagrange s theorem
MA554 Assessment 1 Cosets and Lagrange s theorem These are notes on cosets and Lagrange s theorem; they go over some material from the lectures again, and they have some new material it is all examinable,
More informationA. (Groups of order 8.) (a) Which of the five groups G (as specified in the question) have the following property: G has a normal subgroup N such that
MATH 402A  Solutions for the suggested problems. A. (Groups of order 8. (a Which of the five groups G (as specified in the question have the following property: G has a normal subgroup N such that N =
More informationTHE FUNDAMENTAL GROUP AND CW COMPLEXES
THE FUNDAMENTAL GROUP AND CW COMPLEXES JAE HYUNG SIM Abstract. This paper is a quick introduction to some basic concepts in Algebraic Topology. We start by defining homotopy and delving into the Fundamental
More informationLecture 4. Algebra. Section 1:. Signature, algebra in a signature. Isomorphisms, homomorphisms, congruences and quotient algebras.
V. Borschev and B. Partee, September 18, 2001 p. 1 Lecture 4. Algebra. Section 1:. Signature, algebra in a signature. Isomorphisms, homomorphisms, congruences and quotient algebras. CONTENTS 0. Why algebra?...1
More information