EXAMPLES CLASS 2 MORE COSETS, FIRST INTRODUCTION TO FACTOR GROUPS
|
|
- Eugene Atkins
- 6 years ago
- Views:
Transcription
1 EXAMPLES CLASS 2 MORE COSETS, FIRST INTRODUCTION TO FACTOR GROUPS Let (G, ) be a group, H G, and [G : H] the set of right cosets of H in G. We define a new binary operation on [G : H] by (1) (Hx 1 ) (Hx 2 ) = H(x 1 x 2 ). This may look familiar to some of you: when H is a normal subgroup of G, this is exactly the binary operation on the factor group G/H. Since we haven t defined normal subgroups yet, the point of these questions is to get a feel for what it means to define a binary operation on cosets, and to see what can go wrong when H is not a normal subgroup. 1. G = (Z, +), H = nz, n Questions (a) If you didn t do this last week, work out the cosets for H in G (for a fixed n). Note that they have the form H + x rather than Hx since the binary operation on G is +. (b) Translating the definition in equation (1) to this setting, we have (H + x 1 ) (H + x 2 ) := H + (x 1 + x 2 ). Apply this definition to the cosets you ve worked out in (a). How does this compare to the group (Z n, +) that you re familiar with? 2. G = D 3, H = r G. (a) Write down the (right) cosets of H in G. (b) With the binary operation on [G : H] defined in (1), what group is [G : H] isomorphic to? 3. G = D 3, H = {e, s}. Find an example to show that the multiplication of cosets (1) is not well defined on [G : H], i.e. cosets Hx 1 = Hx 2 and Hy 1 = Hy 2 such that (Hx 1 ) (Hx 2 ) (Hy 1 ) (Hy 2 ). 1
2 2 EXAMPLE CLASS 2 2. Hints To work out the right cosets, follow the same pattern as in Example Class 1: find an element of G that hasn t appeared in any cosets so far, work out its coset, and keep doing this until you ve written down the whole group. Remember that the cosets partition G (viewed as a set rather than a group), so all we re doing is breaking up G into smaller pieces in a well-defined way. In 2(b), it may help to think about the order of [G : H]: we know G : H = G / H, and there aren t many different groups of this size (up to isomorphism). It s a lot easier to write down an isomorphism once you know what group you re trying to identify [G : H] with!
3 1. G = (Z, +), H = nz, n 2. EXAMPLE CLASS Answers (a) If you didn t do this last week, work out the cosets for H in G (for a fixed n). Note that they have the form H + x rather than Hx since the binary operation on G is +. (b) Translating the definition in equation (1) to this setting, we have (H + x 1 ) (H + x 2 ) := H + (x 1 + x 2 ). Apply this definition to the cosets you ve worked out in (a). How does this compare to the group (Z n, +) that you re familiar with? Answer: See the writeup of Example Class 1 for a description of the cosets of nz in Z. They are: nz + 0 = {..., n, 0, n, 2n,... } nz + 1 = {..., n + 1, 1, n + 1, 2n + 1,... }. nz + (n 1) = {..., 1, n 1, 2n 1, 3n 1,... } The transversal set is a set T consisting of exactly one element from each coset. Here we have chosen T = {0, 1, 2,..., n 1} but there are many other choices. The chosen element from each coset is representing that coset: in this example, since H = nz consists of multiples of n, a coset nz + x consists of all integers which give a remainder of x when divided by n. So 1 and n + 1 represent the same coset because they tell us the same piece of information: all elements in nz + 1 = nz + (n + 1) give a remainder of 1 when divided by n. We decide that we might as well think of 1 and n + 1 as equal here, because their cosets are equal. Now consider [Z : nz], the set of cosets of nz in Z (here we don t need to specify right/left cosets because Z is abelian) and define the binary operation (1) on it. We ll take T = {0, 1, 2,..., n 1} for simplicity. If nz + x, nz + y are two cosets in [Z : nz], then their sum is defined to be the coset nz + (x + y). This makes sense: for any an + x nz + x, any bn+y nz+y, their sum is an+x+bn+y = (a+b)n+(x+y) nz+(x+y). What is nz + (x + y)? If (x + y) T then we will leave it in that form; otherwise, x + y n and we have an equality of cosets nz + (x + y) = nz + (x + y n). All we have done is reduced x + y modulo n. In other words, the set of cosets [Z : nz] with the binary operation in (1) behaves in exactly the same way as (Z n, +), except that the elements of [Z : nz] are cosets and the elements of Z n are integers. In fact, this is often
4 4 EXAMPLE CLASS 2 how Z n is defined: we fix the transversal T = {0, 1, 2,..., n 1} for nz in Z and write x to mean the coset nz + x. So if you re happy working with integers modulo n, you ve already mastered one example of working with cosets! 2. G = D 3, H = r G. (a) Write down the (right) cosets of H in G. (b) With the binary operation on [G : H] defined in (1), what group is [G : H] isomorphic to? Answer: Again, see Example Class 1 for the process of working out the cosets. We ll work with right cosets here, but we ll see in Week 4 that H is a normal subgroup of G, which means that the left cosets and right cosets of H in G are equal. We have Hr = {e, r, r 2 } Hs = {s, rs, r 2 s} Here we re working with the transversal T = {r, s}, although again there are many other choices. The following approach may help you understand the group structure on the cosets: we are essentially defining a group structure on the elements of T using the binary operation on G, and we force this binary operation to be closed on T by declaring that anything in the coset Hx is equal to x, for each x T. The binary operation on the set of cosets [G : H] is Hx 1 Hx 2 = H(x 1 x 2 ), where the product x 1 x 2 on the RHS is the binary operation in G. In our example, we have (Hr) (Hr) = H(r 2 ) = Hr, (Hr) (Hs) = H(rs) = Hs, (Hs) (Hr) = H(sr) = Hs, (Hs) (Hs) = H(s 2 ) = Hr, since Hr = Hr 2 as cosets, since Hrs = Hs as cosets, since sr = r 1 s = r 2 s and Hr 2 s = Hs, since s 2 = e and He = Hr. Equally, we could have declared that from now on we were going to assume that e = r = r 2 and s = rs = r 2 s, and worked out what structure this defined on our transversal set T = {r, s} when we multiply them using the binary operation from G. We get r r = r, r s = s, s r = s, s s = r, which is exactly what we had in the coset computation above as well. Notice that r is acting as the identity in this new group (which makes sense, it s in the same coset as e), and s is the only non-trivial element. There is only one group with this property: (Z 2, +). We obtain an isomorphism [D 3 : r ] = (Z 2, +).
5 EXAMPLE CLASS G = D 3, H = {e, s}. Find an example to show that the multiplication of cosets (1) is not well defined on [G : H], i.e. cosets Hx 1 = Hx 2 and Hy 1 = Hy 2 such that (Hx 1 ) (Hx 2 ) (Hy 1 ) (Hy 2 ). Answer: This question illustrates why we only ever consider the binary operation (1) when H is a normal subgroup of G. The right cosets of H in G are He = {e, s} Hr = {r, sr} Hr 2 = {r 2, sr 2 } We have Hsr = Hr and Hr 2 = Hsr 2, since any element of a coset represents that coset. However, if we try to multiply them: (Hr 2 ) (Hsr) = H(r 2 sr) = H(r 1 sr) = H(sr 2 ) (Hsr 2 ) (Hr) = H(sr 3 ) = Hs But Hsr 2 and Hs are completely different cosets! The binary operation from (1) is not well defined, because we get different answers to the same question depending on how we look at it. Thinking about this in the same terms as question 2: we ve tried to declare that e = s, r = sr, and r 2 = sr 2 (and no other equalities), so we would expect that r 2 sr = sr 2 r. But r 2 sr = sr 2, while sr 2 r = s. We decided that sr 2 and s were not equal to each other, which means that our multiplication does not make sense.
6 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 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 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 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 informationCosets, Lagrange s Theorem, and Normal Subgroups
Chapter 7 Cosets, Lagrange s Theorem, and Normal Subgroups 7.1 Cosets Undoubtably, you ve noticed numerous times that if G is a group with H apple G and g 2 G, then both H and g divide G. The theorem that
More informationCosets, Lagrange s Theorem, and Normal Subgroups
Chapter 7 Cosets, Lagrange s Theorem, and Normal Subgroups 7.1 Cosets Undoubtably, you ve noticed numerous times that if G is a group with H apple G and g 2 G, then both H and g divide G. The theorem that
More informationMath101, Sections 2 and 3, Spring 2008 Review Sheet for Exam #2:
Math101, Sections 2 and 3, Spring 2008 Review Sheet for Exam #2: 03 17 08 3 All about lines 3.1 The Rectangular Coordinate System Know how to plot points in the rectangular coordinate system. Know the
More information2MA105 Algebraic Structures I
2MA105 Algebraic Structures I Per-Anders 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 informationMATH EXAMPLES: GROUPS, SUBGROUPS, COSETS
MATH 370 - EXAMPLES: GROUPS, SUBGROUPS, COSETS DR. ZACHARY SCHERR There seemed to be a lot of confusion centering around cosets and subgroups generated by elements. The purpose of this document is to supply
More information7. Let K = 15 be the subgroup of G = Z generated by 15. (a) List the elements of K = 15. Answer: K = 15 = {15k k Z} (b) Prove that K is normal subgroup of G. Proof: (Z +) is Abelian group and any subgroup
More informationAlex s Guide to Word Problems and Linear Equations Following Glencoe Algebra 1
Alex s Guide to Word Problems and Linear Equations Following Glencoe Algebra 1 What is a linear equation? It sounds fancy, but linear equation means the same thing as a line. In other words, it s an equation
More information5.1 Simplifying Rational Expressions
5. Simplifying Rational Expressions Now that we have mastered the process of factoring, in this chapter, we will have to use a great deal of the factoring concepts that we just learned. We begin with the
More informationChapter 5. A Formal Approach to Groups. 5.1 Binary Operations
Chapter 5 A Formal Approach to Groups In this chapter we finally introduce the formal definition of a group. From this point on, our focus will shift from developing intuition to studying the abstract
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 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 informationAlgebra 1B notes and problems March 12, 2009 Factoring page 1
March 12, 2009 Factoring page 1 Factoring Last class, you worked on a set of problems where you had to do multiplication table calculations in reverse. For example, given the answer x 2 + 4x + 2x + 8,
More informationMATH 25 CLASS 21 NOTES, NOV Contents. 2. Subgroups 2 3. Isomorphisms 4
MATH 25 CLASS 21 NOTES, NOV 7 2011 Contents 1. Groups: definition 1 2. Subgroups 2 3. Isomorphisms 4 1. Groups: definition Even though we have been learning number theory without using any other parts
More informationCosets and Lagrange s theorem
Cosets and Lagrange s theorem These are notes on cosets and Lagrange s theorem some of which may already have been lecturer. There are some questions for you included in the text. You should write the
More informationbase 2 4 The EXPONENT tells you how many times to write the base as a factor. Evaluate the following expressions in standard notation.
EXPONENTIALS Exponential is a number written with an exponent. The rules for exponents make computing with very large or very small numbers easier. Students will come across exponentials in geometric sequences
More information#29: Logarithm review May 16, 2009
#29: Logarithm review May 16, 2009 This week we re going to spend some time reviewing. I say re- view since you ve probably seen them before in theory, but if my experience is any guide, it s quite likely
More informationCosets. gh = {gh h H}. Hg = {hg h H}.
Cosets 10-4-2006 If H is a subgroup of a group G, a left coset of H in G is a subset of the form gh = {gh h H}. A right coset of H in G is a subset of the form Hg = {hg h H}. The collection of left cosets
More informationQuotient Rings. is defined. Addition of cosets is defined by adding coset representatives:
Quotient Rings 4-21-2018 Let R be a ring, and let I be a (two-sided) ideal. Considering just the operation of addition, R is a group and I is a subgroup. In fact, since R is an abelian group under addition,
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 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 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 informationSolving Quadratic & Higher Degree Equations
Chapter 7 Solving Quadratic & Higher Degree Equations Sec 1. Zero Product Property Back in the third grade students were taught when they multiplied a number by zero, the product would be zero. In algebra,
More informationMath 280 Modern Algebra Assignment 3 Solutions
Math 280 Modern Algebra Assignment 3 s 1. Below is a list of binary operations on a given set. Decide if each operation is closed, associative, or commutative. Justify your answers in each case; if an
More informationSolving Quadratic & Higher Degree Equations
Chapter 9 Solving Quadratic & Higher Degree Equations Sec 1. Zero Product Property Back in the third grade students were taught when they multiplied a number by zero, the product would be zero. In algebra,
More informationLecture 2: Groups. Rajat Mittal. IIT Kanpur
Lecture 2: Groups Rajat Mittal IIT Kanpur These notes are about the first abstract mathematical structure we are going to study, groups. You are already familiar with set, which is just a collection of
More informationQ 2.0.2: If it s 5:30pm now, what time will it be in 4753 hours? Q 2.0.3: Today is Wednesday. What day of the week will it be in one year from today?
2 Mod math Modular arithmetic is the math you do when you talk about time on a clock. For example, if it s 9 o clock right now, then it ll be 1 o clock in 4 hours. Clearly, 9 + 4 1 in general. But on a
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 informationEquivalence Relations and Partitions, Normal Subgroups, Quotient Groups, and Homomorphisms
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
More informationInduction and Mackey Theory
Induction and Mackey Theory I m writing this short handout to try and explain what the idea of Mackey theory is. The aim of this is not to replace proofs/definitions in the lecture notes, but rather to
More informationGetting Started with Communications Engineering. Rows first, columns second. Remember that. R then C. 1
1 Rows first, columns second. Remember that. R then C. 1 A matrix is a set of real or complex numbers arranged in a rectangular array. They can be any size and shape (provided they are rectangular). A
More informationIsomorphisms and Well-definedness
Isomorphisms and Well-definedness Jonathan Love October 30, 2016 Suppose you want to show that two groups G and H are isomorphic. There are a couple of ways to go about doing this depending on the situation,
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 2
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 2 RAVI VAKIL CONTENTS 1. Where we were 1 2. Yoneda s lemma 2 3. Limits and colimits 6 4. Adjoints 8 First, some bureaucratic details. We will move to 380-F for Monday
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 informationSection 15 Factor-group computation and simple groups
Section 15 Factor-group computation and simple groups Instructor: Yifan Yang Fall 2006 Outline Factor-group computation Simple groups The problem Problem Given a factor group G/H, find an isomorphic group
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 informationElementary Properties of the Integers
Elementary Properties of the Integers 1 1. Basis Representation Theorem (Thm 1-3) 2. Euclid s Division Lemma (Thm 2-1) 3. Greatest Common Divisor 4. Properties of Prime Numbers 5. Fundamental Theorem of
More information8 Extension theory. Notes 8 Autumn 2011
MTHM024/MTH714U Group Theory Notes 8 Autumn 2011 8 Extension theory In this section we tackle the harder problem of describing all extensions of a group A by a group H; that is, all groups G which have
More informationMA 1128: Lecture 08 03/02/2018. Linear Equations from Graphs And Linear Inequalities
MA 1128: Lecture 08 03/02/2018 Linear Equations from Graphs And Linear Inequalities Linear Equations from Graphs Given a line, we would like to be able to come up with an equation for it. I ll go over
More informationExercises MAT2200 spring 2013 Ark 4 Homomorphisms and factor groups
Exercises MAT2200 spring 2013 Ark 4 Homomorphisms and factor groups This Ark concerns the weeks No. (Mar ) and No. (Mar ). Plans until Eastern vacations: In the book the group theory included in the curriculum
More informationLecture 6: Finite Fields
CCS Discrete Math I Professor: Padraic Bartlett Lecture 6: Finite Fields Week 6 UCSB 2014 It ain t what they call you, it s what you answer to. W. C. Fields 1 Fields In the next two weeks, we re going
More information5.2 Infinite Series Brian E. Veitch
5. Infinite Series Since many quantities show up that cannot be computed exactly, we need some way of representing it (or approximating it). One way is to sum an infinite series. Recall that a n is the
More informationAlgebra 8.6 Simple Equations
Algebra 8.6 Simple Equations 1. Introduction Let s talk about the truth: 2 = 2 This is a true statement What else can we say about 2 that is true? Eample 1 2 = 2 1+ 1= 2 2 1= 2 4 1 = 2 2 4 2 = 2 4 = 4
More informationElementary Algebra Chinese Remainder Theorem Euclidean Algorithm
Elementary Algebra Chinese Remainder Theorem Euclidean Algorithm April 11, 2010 1 Algebra We start by discussing algebraic structures and their properties. This is presented in more depth than what we
More informationMATH 3030, Abstract Algebra FALL 2012 Toby Kenney Midyear Examination Friday 7th December: 7:00-10:00 PM
MATH 3030, Abstract Algebra FALL 2012 Toby Kenney Midyear Examination Friday 7th December: 7:00-10:00 PM Basic Questions 1. Compute the factor group Z 3 Z 9 / (1, 6). The subgroup generated by (1, 6) is
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 information2. If the values for f(x) can be made as close as we like to L by choosing arbitrarily large. lim
Limits at Infinity and Horizontal Asymptotes As we prepare to practice graphing functions, we should consider one last piece of information about a function that will be helpful in drawing its graph the
More informationHomework 6 Solutions to Selected Problems
Homework 6 Solutions to Selected Problems March 16, 2012 1 Chapter 7, Problem 6 (not graded) Note that H = {bn : b Z}. That is, H is the subgroup of multiples of n. To nd cosets, we look for an integer
More informationMath 31 Lesson Plan. Day 5: Intro to Groups. Elizabeth Gillaspy. September 28, 2011
Math 31 Lesson Plan Day 5: Intro to Groups Elizabeth Gillaspy September 28, 2011 Supplies needed: Sign in sheet Goals for students: Students will: Improve the clarity of their proof-writing. Gain confidence
More informationCommutative Rings and Fields
Commutative Rings and Fields 1-22-2017 Different algebraic systems are used in linear algebra. The most important are commutative rings with identity and fields. Definition. A ring is a set R with two
More informationElectron Counting. It s easier to show than to explain, I think.
Electron Counting This, for some reason I ve never worked out, isn t as easy as it should be. It is, however, important. It allows us to say things about the stability of organometallic complexes and make
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 informationWhen we use asymptotic notation within an expression, the asymptotic notation is shorthand for an unspecified function satisfying the relation:
CS 124 Section #1 Big-Oh, the Master Theorem, and MergeSort 1/29/2018 1 Big-Oh Notation 1.1 Definition Big-Oh notation is a way to describe the rate of growth of functions. In CS, we use it to describe
More informationLesson 6-1: Relations and Functions
I ll bet you think numbers are pretty boring, don t you? I ll bet you think numbers have no life. For instance, numbers don t have relationships do they? And if you had no relationships, life would be
More informationElliptic Curves and Public Key Cryptography
Elliptic Curves and Public Key Cryptography Jeff Achter January 7, 2011 1 Introduction to Elliptic Curves 1.1 Diophantine equations Many classical problems in number theory have the following form: Let
More informationThe group (Z/nZ) February 17, In these notes we figure out the structure of the unit group (Z/nZ) where n > 1 is an integer.
The group (Z/nZ) February 17, 2016 1 Introduction In these notes we figure out the structure of the unit group (Z/nZ) where n > 1 is an integer. If we factor n = p e 1 1 pe, where the p i s are distinct
More informationQuadratic Equations Part I
Quadratic Equations Part I Before proceeding with this section we should note that the topic of solving quadratic equations will be covered in two sections. This is done for the benefit of those viewing
More informationThe PROMYS Math Circle Problem of the Week #3 February 3, 2017
The PROMYS Math Circle Problem of the Week #3 February 3, 2017 You can use rods of positive integer lengths to build trains that all have a common length. For instance, a train of length 12 is a row of
More informationSolving a Series. Carmen Bruni
A Sample Series Problem Question: Does the following series converge or diverge? n=1 n 3 + 3n 2 + 1 n 5 + 14n 3 + 4n First Attempt First let s think about what this series is - maybe the terms are big
More informationKnots, Coloring and Applications
Knots, Coloring and Applications Ben Webster University of Virginia March 10, 2015 Ben Webster (UVA) Knots, Coloring and Applications March 10, 2015 1 / 14 This talk is online at http://people.virginia.edu/~btw4e/knots.pdf
More informationSection 20: Arrow Diagrams on the Integers
Section 0: Arrow Diagrams on the Integers Most of the material we have discussed so far concerns the idea and representations of functions. A function is a relationship between a set of inputs (the leave
More informationMath 31 Lesson Plan. Day 22: Tying Up Loose Ends. Elizabeth Gillaspy. October 31, Supplies needed: Colored chalk.
Math 31 Lesson Plan Day 22: Tying Up Loose Ends Elizabeth Gillaspy October 31, 2011 Supplies needed: Colored chalk Other topics V 4 via (P ({1, 2}), ) and Cayley table. D n for general n; what s the center?
More informationMath Introduction to Modern Algebra
Math 343 - Introduction to Modern Algebra Notes Rings and Special Kinds of Rings Let R be a (nonempty) set. R is a ring if there are two binary operations + and such that (A) (R, +) is an abelian group.
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 informationAlgebra. Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.
This document was written and copyrighted by Paul Dawkins. Use of this document and its online version is governed by the Terms and Conditions of Use located at. The online version of this document is
More informationGroup Isomorphisms - Some Intuition
Group Isomorphisms - Some Intuition The idea of an isomorphism is central to algebra. It s our version of equality - two objects are considered isomorphic if they are essentially the same. Before studying
More informationPolynomial and Synthetic Division
Polynomial and Synthetic Division Polynomial Division Polynomial Division is very similar to long division. Example: 3x 3 5x 3x 10x 1 3 Polynomial Division 3x 1 x 3x 3 3 x 5x 3x x 6x 4 10x 10x 7 3 x 1
More informationIOP2601. Some notes on basic mathematical calculations
IOP601 Some notes on basic mathematical calculations The order of calculations In order to perform the calculations required in this module, there are a few steps that you need to complete. Step 1: Choose
More informationAlgebra Exam. Solutions and Grading Guide
Algebra Exam Solutions and Grading Guide You should use this grading guide to carefully grade your own exam, trying to be as objective as possible about what score the TAs would give your responses. Full
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 non-zero polynomials in [x], no two
More information#26: Number Theory, Part I: Divisibility
#26: Number Theory, Part I: Divisibility and Primality April 25, 2009 This week, we will spend some time studying the basics of number theory, which is essentially the study of the natural numbers (0,
More information1 Introduction. 2 Solving Linear Equations
1 Introduction This essay introduces some new sets of numbers. Up to now, the only sets of numbers (algebraic systems) we know is the set Z of integers with the two operations + and and the system R of
More informationCH 24 IDENTITIES. [Each product is 35] Ch 24 Identities. Introduction
139 CH 4 IDENTITIES Introduction First we need to recall that there are many ways to indicate multiplication; for eample the product of 5 and 7 can be written in a variety of ways: 5 7 5 7 5(7) (5)7 (5)(7)
More information2. l = 7 ft w = 4 ft h = 2.8 ft V = Find the Area of a trapezoid when the bases and height are given. Formula is A = B = 21 b = 11 h = 3 A=
95 Section.1 Exercises Part A Find the Volume of a rectangular solid when the width, height and length are given. Formula is V=lwh 1. l = 4 in w = 2.5 in h = in V = 2. l = 7 ft w = 4 ft h = 2.8 ft V =.
More informationPlease bring the task to your first physics lesson and hand it to the teacher.
Pre-enrolment task for 2014 entry Physics Why do I need to complete a pre-enrolment task? This bridging pack serves a number of purposes. It gives you practice in some of the important skills you will
More informationDeterminants and Scalar Multiplication
Properties of Determinants In the last section, we saw how determinants interact with the elementary row operations. There are other operations on matrices, though, such as scalar multiplication, matrix
More informationWritten Homework # 2 Solution
Math 516 Fall 2006 Radford Written Homework # 2 Solution 10/09/06 Let G be a non-empty set with binary operation. For non-empty subsets S, T G we define the product of the sets S and T by If S = {s} is
More informationGMAT Arithmetic: Challenge (Excerpt)
GMAT Arithmetic: Challenge (Excerpt) Jeff Sackmann / GMAT HACKS January 2013 Contents 1 Introduction 2 2 Difficulty Levels 3 3 Problem Solving 4 4 Data Sufficiency 5 5 Answer Key 7 6 Explanations 8 1 1
More informationSequences and Series
Sequences and Series What do you think of when you read the title of our next unit? In case your answers are leading us off track, let's review the following IB problems. 1 November 2013 HL 2 3 November
More informationExpansion of Terms. f (x) = x 2 6x + 9 = (x 3) 2 = 0. x 3 = 0
Expansion of Terms So, let s say we have a factorized equation. Wait, what s a factorized equation? A factorized equation is an equation which has been simplified into brackets (or otherwise) to make analyzing
More informationWhen we use asymptotic notation within an expression, the asymptotic notation is shorthand for an unspecified function satisfying the relation:
CS 124 Section #1 Big-Oh, the Master Theorem, and MergeSort 1/29/2018 1 Big-Oh Notation 1.1 Definition Big-Oh notation is a way to describe the rate of growth of functions. In CS, we use it to describe
More informationLine Integrals and Path Independence
Line Integrals and Path Independence We get to talk about integrals that are the areas under a line in three (or more) dimensional space. These are called, strangely enough, line integrals. Figure 11.1
More informationMath 131 notes. Jason Riedy. 6 October, Linear Diophantine equations : Likely delayed 6
Math 131 notes Jason Riedy 6 October, 2008 Contents 1 Modular arithmetic 2 2 Divisibility rules 3 3 Greatest common divisor 4 4 Least common multiple 4 5 Euclidean GCD algorithm 5 6 Linear Diophantine
More informationLecture 4: Orbits. Rajat Mittal. IIT Kanpur
Lecture 4: Orbits Rajat Mittal IIT Kanpur In the beginning of the course we asked a question. How many different necklaces can we form using 2 black beads and 10 white beads? In the question, the numbers
More informationSTATISTICS 1 REVISION NOTES
STATISTICS 1 REVISION NOTES Statistical Model Representing and summarising Sample Data Key words: Quantitative Data This is data in NUMERICAL FORM such as shoe size, height etc. Qualitative Data This is
More informationPartial Fractions. June 27, In this section, we will learn to integrate another class of functions: the rational functions.
Partial Fractions June 7, 04 In this section, we will learn to integrate another class of functions: the rational functions. Definition. A rational function is a fraction of two polynomials. For example,
More informationCS173 Strong Induction and Functions. Tandy Warnow
CS173 Strong Induction and Functions Tandy Warnow CS 173 Introduction to Strong Induction (also Functions) Tandy Warnow Preview of the class today What are functions? Weak induction Strong induction A
More informationPreptests 59 Answers and Explanations (By Ivy Global) Section 1 Analytical Reasoning
Preptests 59 Answers and Explanations (By ) Section 1 Analytical Reasoning Questions 1 5 Since L occupies its own floor, the remaining two must have H in the upper and I in the lower. P and T also need
More informationQuiz 07a. Integers Modulo 12
MA 3260 Lecture 07 - Binary Operations Friday, September 28, 2018. Objectives: Continue with binary operations. Quiz 07a We have a machine that is set to run for x hours, turn itself off for 3 hours, and
More information( )( b + c) = ab + ac, but it can also be ( )( a) = ba + ca. Let s use the distributive property on a couple of
Factoring Review for Algebra II The saddest thing about not doing well in Algebra II is that almost any math teacher can tell you going into it what s going to trip you up. One of the first things they
More informationELG 5372 Error Control Coding. Lecture 12: Ideals in Rings and Algebraic Description of Cyclic Codes
ELG 5372 Error Control Coding Lecture 12: Ideals in Rings and Algebraic Description of Cyclic Codes Quotient Ring Example + Quotient Ring Example Quotient Ring Recall the quotient ring R={,,, }, where
More informationCHMC: Finite Fields 9/23/17
CHMC: Finite Fields 9/23/17 1 Introduction This worksheet is an introduction to the fascinating subject of finite fields. Finite fields have many important applications in coding theory and cryptography,
More information10.4 Comparison Tests
0.4 Comparison Tests The Statement Theorem Let a n be a series with no negative terms. (a) a n converges if there is a convergent series c n with a n c n n > N, N Z (b) a n diverges if there is a divergent
More informationGrade 11/12 Math Circles Rational Points on an Elliptic Curves Dr. Carmen Bruni November 11, Lest We Forget
Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 11/12 Math Circles Rational Points on an Elliptic Curves Dr. Carmen Bruni November 11, 2015 - Lest
More informationDivisibility = 16, = 9, = 2, = 5. (Negative!)
Divisibility 1-17-2018 You probably know that division can be defined in terms of multiplication. If m and n are integers, m divides n if n = mk for some integer k. In this section, I ll look at properties
More informationChapters 4/5 Class Notes. Intermediate Algebra, MAT1033C. SI Leader Joe Brownlee. Palm Beach State College
Chapters 4/5 Class Notes Intermediate Algebra, MAT1033C Palm Beach State College Class Notes 4.1 Professor Burkett 4.1 Systems of Linear Equations in Two Variables A system of equations is a set of two
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 information