# Abstract Algebra FINAL EXAM May 23, Name: R. Hammack Score:

Size: px
Start display at page:

## Transcription

1 Abstract Algebra FINAL EXAM May 23, 2003 Name: R. Hammack Score: Directions: Please answer the questions in the space provided. To get full credit you must show all of your work. Use of calculators and other computing or communication devices is not allowed on this test. 1. Draw the subgroup lattice for Z 18. Z List the elements of the cyclic subgroup -i of C. Answer: 1, i, 1, i 3. Find the order of the largest cyclic subgroup of the symmetric group S 10. Consider the element (1,2,3,4,5)(6,7,8)(9,10). It has order (5)(3)(2) = 30, so the subgroup generated by it has 30 elements. Can you do better than this? Any permutation in S 10 can be written as a product of disjoint cycles, and its order is at most the sum of the lengths of the cycles. A quick exhaustive search confrims that the above element has the greatest possible order.

2 4. Consider the set H = {σ S 5 σ(3)=3}. (a) H = 4! = 24 (b) Explain why H is a subgroup of S 5. Note that 1. H is closed. If π, µ H, then π(3)= 3 and µ(3)= 3. Thus πµ(3) = π(µ(3)) = π(3) = 3, so πµ H. 2. The identity permutatio i is in H because i(3) = If µ H, then 3 = µ(3), so µ 1 (3) =µ 1 µ((3))= 3, which means µ 1 is in H. It follows that H is a subgroup. (c) Is H a normal subgroup of S 5? Explain. NO. For example, look at the cycle (1,2,4), which is in H because it leaves 3 unchanged. Cosnsider the permutation (1,3) which is its own inverse. Notice that (1,3)(1,2,4)(1,3) is NOT in H because it sends 3 to 2. This shows that its not true that g 1 hg is H for every element h in H, so H is not normal. (d) How many left cosets of H are there in S 5? There are S 5 / H = 120/24 = 5 such cosets. 5. List all the nonisomorphic groups of order = Z 4 Z 9 Z 5 Z 2 Z 2 Z 9 Z 5 Z 4 Z 3 Z 3 Z 5 Z 2 Z 2 Z 3 Z 3 Z 5 6. Find the order of (3,6,9) in Z 4 Z 12 Z 15. Look at n(3, 6, 9) = (n3, n6, n9), where n is an integer. n must be a multible of 4 to make n3 = 0 n must be a multible of 2 to make n6 = 0 n must be a multible of 5 to make n9 = 0 The least common multiple is 20, so that is the order of (3, 6, 9).

3 7. Are the groups Z 8 Z 10 Z 3 and Z 8 Z 2 Z 15 isomorphic? Why or why not? Z 8 Z 10 Z 3 = Z 8 Z 30 (since 3 and 10 are relatively prime) Z 8 Z 2 Z 15 = Z 8 Z 30 (since 2 and 15 are relatively prime) Therefore the two groups are isomorphic. 8. Find the kernel of the homomorphism φ:z Z 8 for which φ(1)=6. Note φ(n) = φ( ) = φ(1) + φ(1) φ(1) = = 6n (mod 8) Thus the kernel will be all integers n for which 6n = (3)(2)n is a multiple of 8. Such an n must be a multiple of 4. Thus kernel is 4Z. 9. Find the kernel of the homomorphism φ:z 40 Z 5 Z 8 for which φ(1)=(1,4). Note φ(n) = φ( ) = φ(1) + φ(1) φ(1) = n(1, 4) = (n, 4n) For this equal (0,0), n must be a multiple of 5 and 4n must be a multiple of 8. It follows that the kernel is {0, 10, 20, 30} 10. (a) List the units in the ring Z 12. 1, 5, 7, 11 (b) List the zero divisors in the ring Z 12. 2, 3, 4, 6, 8, 9, 10 (c) List the prime ideals in the ring Z 12. Recall that an ideal N is prime if and olly if Z 12 /N is an integral domain. The ideals in this ring are 0, 1 = 5 = 7 = 11, 2 = 10, 3 = 9, 6, 4 = 8. Z 12 / 0 = Z 12 is not an integral domain so 0 is not prime. Z 12 / 1 = {0} is not an integral domain so 1 is not prime. Z 12 / 2 = Z 2 is an integral domain so 2 is prime. Z 12 / 3 = Z 4 is not an integral domain so 3 is not prime. Z 12 / 4 = Z 3 is an integral domain so 4 is prime. Z 12 / 6 = Z 6 is not an integral domain so 6 is not prime. Prime ideals are 2 and 4.

4 11. What familiar group is (Z 4 Z 6 )/ (2,3) isomorphic to? Note H = (2, 3) = {(0,0), (2,3)} has just 2 elements. It follows that the factor group has (4)(6)/2 = 12 elements. We claim that the factor group is generated by the element (1, 1)+ H. 0((1, 1)+ H ) = (0, 0)+ H 1((1, 1)+ H ) = (1, 1)+ H 2((1, 1)+ H ) = (2, 2)+ H 3((1, 1)+ H ) = (3, 3)+ H 4((1, 1)+ H ) = (0, 4)+ H 5((1, 1)+ H ) = (1, 5)+ H 6((1, 1)+ H ) = (2, 0)+ H 7((1, 1)+ H ) = (3, 1)+ H 8((1, 1)+ H ) = (0, 2)+ H 9((1, 1)+ H ) = (1, 3)+ H 10((1, 1)+ H ) = (2, 4)+ H 11((1, 1)+ H ) = (3, 5)+ H 12((1, 1)+ H ) = (0, 0)+ H < finally "cycles" back to the identity here. Thus (1, 1)+ H generates the entire group. Group is cyclic with 12 elements. It s Z Explain why C /U R +. Consider the function φ : C R +, given by φ(z) = z. This is a homomorphism becuase φ(zw) = zw = z w = φ(z)φ(w). It s surjective because given any x in R +, φ(x) = x. Also, its Kernel is {z C : φ(z) = 1} = {z C : z = 1} = U. By the Fundamental Theorem of Homomorphisms, there is an isomorphism µ : C /U R Is 2x 3 +x 2 + 2x +2 an irreducible polynomial in Z 5 [x]? If not, write it as a product of irreducible polynomials. Let f(x) =2x 3 +x 2 + 2x +2. If this factored, then it would factor into a linear and a quadratic term, or 3 linear terms. Either way, there would be a linear term, so the polynomial would have a root. But a quick check shows there are no roots: f (0) = 2 f (1) = 2 f (2) = 1 f (3) = 1 f (4) = 4 Conclusion. It can t be factored. It s irreducible.

5 14. Find all c Z 3 for which Z 3 [x]/ x 2 +c is a field. These would be all the elements c for which the ideal x 2 + c is maximal, which in turn is all elements c for which x 2 +c is irrecucible. If c = 0, the polynomial is x 2 = (x)(x) which is not irreducible. If c = 1, the polynomial is x 2 +1, and its of degree 2 with no roots, so its irreducible. If c = 2, the polynomial is x 2 +2, and its of degree 2 with no roots, so its irreducible. ANSWER: c = 1 and c = Prove that if G is a finite group with identity e, and m = G, then x m = e for any element x G. Proof. Take any x in G and consider the cyclic subgroup x. Let s say k = x, which means x = {e, x, x 2, x 3, x 4,, x k 1 }, so x k = e. Lagrange s Theorem says k divides m, so m = kn for some integer n. Now, x m = x kn = ( x k) n = e n = e. 16. Suppose that G is a group with identity e. Prove that if x 2 = e for every element x in G, then G is abelian. Proof. Suppose a and b are arbitrary elements of G. We want to show ab = ba. By hypothesis, (ab) 2 = abab = e. Multiply both sides of abab = e on the left by a and you get aabab = a. But, since aa= e, this becomes bab = a. Now multiply both sides of bab = a on the right by b to get babb = ab. But since bb = e this becomes ba = ab. Therefore G is abelian.

6 17. Prove that if G is an abelian group, then the set of all elements x G for which x 2 = e form a subgroup of G. Proof. Let H = {x G x 2 = e}. We must show this is a subgroup of G. Notice that: 1. H is closed. If a, b H, then a 2 = e and b 2 = e, so (ab) 2 = abab = aabb = a 2 b 2 = ee = e, so ab is in H. 2. The identity e is in H because e 2 = e. 3. If a is in H, then a 2 = e so ( a 2) 1 =e 1, which is a 2 = e, or ( a 1) 2 = e. This means a 1 is in H. 18. Prove that the units of a ring with unity form a multiplicative group. Proof. Suppose R is a ring with unity and M R is the set of all its units. Notice that M is closed under multiplication, for if a and b are in M then ab is a unit with inverse b 1 a 1. Thus ring multiplication gives a binary operation on M. We now just need to show the 3 group axioms hold for multiplication in M. 1. Multiplication is assosiative because it s assosiative in the ring R. 2. Unity 1 is in M because it s a unit, and this serves as the identity. 3. If a is in M, then a is a unit and so is its inverse because aa 1 = 1, so a 1 is in M. We re done.

### Name: Solutions Final Exam

Instructions. Answer each of the questions on your own paper. Be sure to show your work so that partial credit can be adequately assessed. Put your name on each page of your paper. 1. [10 Points] All of

### Modern 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

### Basic 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

### Page Points Possible Points. Total 200

Instructions: 1. The point value of each exercise occurs adjacent to the problem. 2. No books or notes or calculators are allowed. Page Points Possible Points 2 20 3 20 4 18 5 18 6 24 7 18 8 24 9 20 10

### Algebraic 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

### Name: Solutions Final Exam

Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Put your name on each page of your paper. 1. [10 Points] For

### MATH 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

### Some practice problems for midterm 2

Some practice problems for midterm 2 Kiumars Kaveh November 14, 2011 Problem: Let Z = {a G ax = xa, x G} be the center of a group G. Prove that Z is a normal subgroup of G. Solution: First we prove Z is

### ALGEBRA PH.D. QUALIFYING EXAM September 27, 2008

ALGEBRA PH.D. QUALIFYING EXAM September 27, 2008 A passing paper consists of four problems solved completely plus significant progress on two other problems; moreover, the set of problems solved completely

### Supplement. 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

### 5 Group theory. 5.1 Binary operations

5 Group theory This section is an introduction to abstract algebra. This is a very useful and important subject for those of you who will continue to study pure mathematics. 5.1 Binary operations 5.1.1

### Definitions. 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

### Definition 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.

### Homework problems from Chapters IV-VI: answers and solutions

Homework problems from Chapters IV-VI: answers and solutions IV.21.1. In this problem we have to describe the field F of quotients of the domain D. Note that by definition, F is the set of equivalence

### B Sc MATHEMATICS ABSTRACT ALGEBRA

UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B Sc MATHEMATICS (0 Admission Onwards) V Semester Core Course ABSTRACT ALGEBRA QUESTION BANK () Which of the following defines a binary operation on Z

### Total 100

Math 542 Midterm Exam, Spring 2016 Prof: Paul Terwilliger Your Name (please print) SOLUTIONS NO CALCULATORS/ELECTRONIC DEVICES ALLOWED. MAKE SURE YOUR CELL PHONE IS OFF. Problem Value 1 10 2 10 3 10 4

### MATH 420 FINAL EXAM J. Beachy, 5/7/97

MATH 420 FINAL EXAM J. Beachy, 5/7/97 1. (a) For positive integers a and b, define gcd(a, b). (b) Compute gcd(1776, 1492). (c) Show that if a, b, c are positive integers, then gcd(a, bc) = 1 if and only

### January 2016 Qualifying Examination

January 2016 Qualifying Examination If you have any difficulty with the wording of the following problems please contact the supervisor immediately. All persons responsible for these problems, in principle,

### Section 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

### Section VI.33. Finite Fields

VI.33 Finite Fields 1 Section VI.33. Finite Fields Note. In this section, finite fields are completely classified. For every prime p and n N, there is exactly one (up to isomorphism) field of order p n,

### Factorization in Polynomial Rings

Factorization in Polynomial Rings Throughout these notes, F denotes a field. 1 Long division with remainder We begin with some basic definitions. Definition 1.1. Let f, g F [x]. We say that f divides g,

### INTRODUCTION TO THE GROUP THEORY

Lecture Notes on Structure of Algebra INTRODUCTION TO THE GROUP THEORY By : Drs. Antonius Cahya Prihandoko, M.App.Sc e-mail: antoniuscp.fkip@unej.ac.id Mathematics Education Study Program Faculty of Teacher

### EXAM 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

### Teddy Einstein Math 4320

Teddy Einstein Math 4320 HW4 Solutions Problem 1: 2.92 An automorphism of a group G is an isomorphism G G. i. Prove that Aut G is a group under composition. Proof. Let f, g Aut G. Then f g is a bijective

### Algebra Ph.D. Entrance Exam Fall 2009 September 3, 2009

Algebra Ph.D. Entrance Exam Fall 2009 September 3, 2009 Directions: Solve 10 of the following problems. Mark which of the problems are to be graded. Without clear indication which problems are to be graded

### Mathematics for Cryptography

Mathematics for Cryptography Douglas R. Stinson David R. Cheriton School of Computer Science University of Waterloo Waterloo, Ontario, N2L 3G1, Canada March 15, 2016 1 Groups and Modular Arithmetic 1.1

### Finite Fields. [Parts from Chapter 16. Also applications of FTGT]

Finite Fields [Parts from Chapter 16. Also applications of FTGT] Lemma [Ch 16, 4.6] Assume F is a finite field. Then the multiplicative group F := F \ {0} is cyclic. Proof Recall from basic group theory

### Math 120. Groups and Rings Midterm Exam (November 8, 2017) 2 Hours

Math 120. Groups and Rings Midterm Exam (November 8, 2017) 2 Hours Name: Please read the questions carefully. You will not be given partial credit on the basis of having misunderstood a question, and please

### PRACTICE FINAL MATH , MIT, SPRING 13. You have three hours. This test is closed book, closed notes, no calculators.

PRACTICE FINAL MATH 18.703, MIT, SPRING 13 You have three hours. This test is closed book, closed notes, no calculators. There are 11 problems, and the total number of points is 180. Show all your work.

### Math 581 Problem Set 8 Solutions

Math 581 Problem Set 8 Solutions 1. Prove that a group G is abelian if and only if the function ϕ : G G given by ϕ(g) g 1 is a homomorphism of groups. In this case show that ϕ is an isomorphism. Proof:

### Midterm Exam. There are 6 problems. Your 5 best answers count. Please pay attention to the presentation of your work! Best 5

Department of Mathematical Sciences Instructor: Daiva Pucinskaite Modern Algebra June 22, 2017 Midterm Exam There are 6 problems. Your 5 best answers count. Please pay attention to the presentation of

### NOTES ON FINITE FIELDS

NOTES ON FINITE FIELDS AARON LANDESMAN CONTENTS 1. Introduction to finite fields 2 2. Definition and constructions of fields 3 2.1. The definition of a field 3 2.2. Constructing field extensions by adjoining

### Elements of solution for Homework 5

Elements of solution for Homework 5 General remarks How to use the First Isomorphism Theorem A standard way to prove statements of the form G/H is isomorphic to Γ is to construct a homomorphism ϕ : G Γ

### Abstract Algebra, Second Edition, by John A. Beachy and William D. Blair. Corrections and clarifications

1 Abstract Algebra, Second Edition, by John A. Beachy and William D. Blair Corrections and clarifications Note: Some corrections were made after the first printing of the text. page 9, line 8 For of the

### MATH 4107 (Prof. Heil) PRACTICE PROBLEMS WITH SOLUTIONS Spring 2018

MATH 4107 (Prof. Heil) PRACTICE PROBLEMS WITH SOLUTIONS Spring 2018 Here are a few practice problems on groups. You should first work through these WITHOUT LOOKING at the solutions! After you write your

### MA441: Algebraic Structures I. Lecture 18

MA441: Algebraic Structures I Lecture 18 5 November 2003 1 Review from Lecture 17: Theorem 6.5: Aut(Z/nZ) U(n) For every positive integer n, Aut(Z/nZ) is isomorphic to U(n). The proof used the map T :

### Galois fields/1. (M3) There is an element 1 (not equal to 0) such that a 1 = a for all a.

Galois fields 1 Fields A field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except by zero) can be performed, and satisfy the usual rules. More

### MATH 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,

### Fall /29/18 Time Limit: 75 Minutes

Math 411: Abstract Algebra Fall 2018 Midterm 10/29/18 Time Limit: 75 Minutes Name (Print): Solutions JHU-ID: This exam contains 8 pages (including this cover page) and 6 problems. Check to see if any pages

### Math 4400, Spring 08, Sample problems Final Exam.

Math 4400, Spring 08, Sample problems Final Exam. 1. Groups (1) (a) Let a be an element of a group G. Define the notions of exponent of a and period of a. (b) Suppose a has a finite period. Prove that

### Abstract Algebra Study Sheet

Abstract Algebra Study Sheet This study sheet should serve as a guide to which sections of Artin will be most relevant to the final exam. When you study, you may find it productive to prioritize the definitions,

### φ(xy) = (xy) n = x n y n = φ(x)φ(y)

Groups 1. (Algebra Comp S03) Let A, B and C be normal subgroups of a group G with A B. If A C = B C and AC = BC then prove that A = B. Let b B. Since b = b1 BC = AC, there are a A and c C such that b =

### 1 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

### Algebra Exam, Spring 2017

Algebra Exam, Spring 2017 There are 5 problems, some with several parts. Easier parts count for less than harder ones, but each part counts. Each part may be assumed in later parts and problems. Unjustified

### Rings. 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

### Solutions of Assignment 10 Basic Algebra I

Solutions of Assignment 10 Basic Algebra I November 25, 2004 Solution of the problem 1. Let a = m, bab 1 = n. Since (bab 1 ) m = (bab 1 )(bab 1 ) (bab 1 ) = ba m b 1 = b1b 1 = 1, we have n m. Conversely,

### Math 546, Exam 2 Information.

Math 546, Exam 2 Information. 10/21/09, LC 303B, 10:10-11:00. Exam 2 will be based on: Sections 3.2, 3.3, 3.4, 3.5; The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/546fa09/546.html)

### 1. (7 points)state and PROVE Cayley s Theorem.

Math 546, Final Exam, Fall 2004 The exam is worth 100 points. Write your answers as legibly as you can on the blank sheets of paper provided. Use only one side of each sheet. Take enough space for each

### Solutions of exercise sheet 4

D-MATH Algebra I HS 14 Prof. Emmanuel Kowalski Solutions of exercise sheet 4 The content of the marked exercises (*) should be known for the exam. 1. Prove the following two properties of groups: 1. Every

### ENTRY 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.

### Chapter 3. Rings. The basic commutative rings in mathematics are the integers Z, the. Examples

Chapter 3 Rings Rings are additive abelian groups with a second operation called multiplication. The connection between the two operations is provided by the distributive law. Assuming the results of Chapter

### DIHEDRAL GROUPS II KEITH CONRAD

DIHEDRAL GROUPS II KEITH CONRAD We will characterize dihedral groups in terms of generators and relations, and describe the subgroups of D n, including the normal subgroups. We will also introduce an infinite

### 1 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

### Ph.D. Qualifying Examination in Algebra Department of Mathematics University of Louisville January 2018

Ph.D. Qualifying Examination in Algebra Department of Mathematics University of Louisville January 2018 Do 6 problems with at least 2 in each section. Group theory problems: (1) Suppose G is a group. The

### 1. Group Theory Permutations.

1.1. Permutations. 1. Group Theory Problem 1.1. Let G be a subgroup of S n of index 2. Show that G = A n. Problem 1.2. Find two elements of S 7 that have the same order but are not conjugate. Let π S 7

### Introduction to finite fields

Chapter 7 Introduction to finite fields This chapter provides an introduction to several kinds of abstract algebraic structures, particularly groups, fields, and polynomials. Our primary interest is in

### Solutions of exercise sheet 8

D-MATH Algebra I HS 14 Prof. Emmanuel Kowalski Solutions of exercise sheet 8 1. In this exercise, we will give a characterization for solvable groups using commutator subgroups. See last semester s (Algebra

Graduate Preliminary Examination Algebra II 18.2.2005: 3 hours Problem 1. Prove or give a counter-example to the following statement: If M/L and L/K are algebraic extensions of fields, then M/K is algebraic.

Answers to Final Exam MA441: Algebraic Structures I 20 December 2003 1) Definitions (20 points) 1. Given a subgroup H G, define the quotient group G/H. (Describe the set and the group operation.) The quotient

### MATH 433 Applied Algebra Lecture 22: Review for Exam 2.

MATH 433 Applied Algebra Lecture 22: Review for Exam 2. Topics for Exam 2 Permutations Cycles, transpositions Cycle decomposition of a permutation Order of a permutation Sign of a permutation Symmetric

### SF2729 GROUPS AND RINGS LECTURE NOTES

SF2729 GROUPS AND RINGS LECTURE NOTES 2011-03-01 MATS BOIJ 6. THE SIXTH LECTURE - GROUP ACTIONS In the sixth lecture we study what happens when groups acts on sets. 1 Recall that we have already when looking

### Homework #5 Solutions

Homework #5 Solutions p 83, #16. In order to find a chain a 1 a 2 a n of subgroups of Z 240 with n as large as possible, we start at the top with a n = 1 so that a n = Z 240. In general, given a i we will

### MATH 113 FINAL EXAM December 14, 2012

p.1 MATH 113 FINAL EXAM December 14, 2012 This exam has 9 problems on 18 pages, including this cover sheet. The only thing you may have out during the exam is one or more writing utensils. You have 180

### and this makes M into an R-module by (1.2). 2

1. Modules Definition 1.1. Let R be a commutative ring. A module over R is set M together with a binary operation, denoted +, which makes M into an abelian group, with 0 as the identity element, together

### I216e Discrete Math (for Review)

I216e Discrete Math (for Review) Nov 22nd, 2017 To check your understanding. Proofs of do not appear in the exam. 1 Monoid Let (G, ) be a monoid. Proposition 1 Uniquness of Identity An idenity e is unique,

### Math 430 Final Exam, Fall 2008

IIT Dept. Applied Mathematics, December 9, 2008 1 PRINT Last name: Signature: First name: Student ID: Math 430 Final Exam, Fall 2008 Grades should be posted Friday 12/12. Have a good break, and don t forget

### Algebra Exam Fall Alexander J. Wertheim Last Updated: October 26, Groups Problem Problem Problem 3...

Algebra Exam Fall 2006 Alexander J. Wertheim Last Updated: October 26, 2017 Contents 1 Groups 2 1.1 Problem 1..................................... 2 1.2 Problem 2..................................... 2

### Supplementary Notes: Simple Groups and Composition Series

18.704 Supplementary Notes: Simple Groups and Composition Series Genevieve Hanlon and Rachel Lee February 23-25, 2005 Simple Groups Definition: A simple group is a group with no proper normal subgroup.

### Solutions I.N. Herstein- Second Edition

Solutions I.N. Herstein- Second Edition Sadiah Zahoor Please email me if any corrections at sadiahzahoor@cantab.net. R is a ring in all problems. Problem 0.1. If a, b, c, d R, evaluate (a + b)(c + d).

### Groups. Groups. 1.Introduction. 1.Introduction. TS.NguyễnViết Đông. 1. Introduction 2.Normal subgroups, quotien groups. 3. Homomorphism.

Groups Groups 1. Introduction 2.Normal sub, quotien. 3. Homomorphism. TS.NguyễnViết Đông 1 2 1.1. Binary Operations 1.2.Definition of Groups 1.3.Examples of Groups 1.4.Sub 1.1. Binary Operations 1.2.Definition

### Algebraic structures I

MTH5100 Assignment 1-10 Algebraic structures I For handing in on various dates January March 2011 1 FUNCTIONS. Say which of the following rules successfully define functions, giving reasons. For each one

### MATH 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

### ALGEBRA I (LECTURE NOTES 2017/2018) LECTURE 9 - CYCLIC GROUPS AND EULER S FUNCTION

ALGEBRA I (LECTURE NOTES 2017/2018) LECTURE 9 - CYCLIC GROUPS AND EULER S FUNCTION PAVEL RŮŽIČKA 9.1. Congruence modulo n. Let us have a closer look at a particular example of a congruence relation on

### Part 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

### Quasi-reducible Polynomials

Quasi-reducible Polynomials Jacques Willekens 06-Dec-2008 Abstract In this article, we investigate polynomials that are irreducible over Q, but are reducible modulo any prime number. 1 Introduction Let

### Groups 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

### Definitions, Theorems and Exercises. Abstract Algebra Math 332. Ethan D. Bloch

Definitions, Theorems and Exercises Abstract Algebra Math 332 Ethan D. Bloch December 26, 2013 ii Contents 1 Binary Operations 3 1.1 Binary Operations............................... 4 1.2 Isomorphic Binary

### Converse to Lagrange s Theorem Groups

Converse to Lagrange s Theorem Groups Blain A Patterson Youngstown State University May 10, 2013 History In 1771 an Italian mathematician named Joseph Lagrange proved a theorem that put constraints on

### Quizzes for Math 401

Quizzes for Math 401 QUIZ 1. a) Let a,b be integers such that λa+µb = 1 for some inetegrs λ,µ. Prove that gcd(a,b) = 1. b) Use Euclid s algorithm to compute gcd(803, 154) and find integers λ,µ such that

### ABSTRACT 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

### Modern Computer Algebra

Modern Computer Algebra Exercises to Chapter 25: Fundamental concepts 11 May 1999 JOACHIM VON ZUR GATHEN and JÜRGEN GERHARD Universität Paderborn 25.1 Show that any subgroup of a group G contains the neutral

### Group. Orders. Review. Orders: example. Subgroups. Theorem: Let x be an element of G. The order of x divides the order of G

Victor Adamchik Danny Sleator Great Theoretical Ideas In Computer Science Algebraic Structures: Group Theory II CS 15-251 Spring 2010 Lecture 17 Mar. 17, 2010 Carnegie Mellon University Group A group G

### Algebra I: Final 2012 June 22, 2012

1 Algebra I: Final 2012 June 22, 2012 Quote the following when necessary. A. Subgroup H of a group G: H G = H G, xy H and x 1 H for all x, y H. B. Order of an Element: Let g be an element of a group G.

### Math 3140 Fall 2012 Assignment #3

Math 3140 Fall 2012 Assignment #3 Due Fri., Sept. 21. Remember to cite your sources, including the people you talk to. My solutions will repeatedly use the following proposition from class: Proposition

### MATH 3030, Abstract Algebra Winter 2012 Toby Kenney Sample Midterm Examination Model Solutions

MATH 3030, Abstract Algebra Winter 2012 Toby Kenney Sample Midterm Examination Model Solutions Basic Questions 1. Give an example of a prime ideal which is not maximal. In the ring Z Z, the ideal {(0,

### List of topics for the preliminary exam in algebra

List of topics for the preliminary exam in algebra 1 Basic concepts 1. Binary relations. Reflexive, symmetric/antisymmetryc, and transitive relations. Order and equivalence relations. Equivalence classes.

### Algebra Qualifying Exam, Fall 2018

Algebra Qualifying Exam, Fall 2018 Name: Student ID: Instructions: Show all work clearly and in order. Use full sentences in your proofs and solutions. All answers count. In this exam, you may use the

### Math 2070BC Term 2 Weeks 1 13 Lecture Notes

Math 2070BC 2017 18 Term 2 Weeks 1 13 Lecture Notes Keywords: group operation multiplication associative identity element inverse commutative abelian group Special Linear Group order infinite order cyclic

### Commutative 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

7. 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

### Johns Hopkins University, Department of Mathematics Abstract Algebra - Spring 2013 Midterm Exam Solution

Johns Hopkins University, Department of Mathematics 110.40 Abstract Algebra - Spring 013 Midterm Exam Solution Instructions: This exam has 6 pages. No calculators, books or notes allowed. You must answer

### Math 210A: Algebra, Homework 5

Math 210A: Algebra, Homework 5 Ian Coley November 5, 2013 Problem 1. Prove that two elements σ and τ in S n are conjugate if and only if type σ = type τ. Suppose first that σ and τ are cycles. Suppose

### Chapter 5. Modular arithmetic. 5.1 The modular ring

Chapter 5 Modular arithmetic 5.1 The modular ring Definition 5.1. Suppose n N and x, y Z. Then we say that x, y are equivalent modulo n, and we write x y mod n if n x y. It is evident that equivalence

### Problem Set Mash 1. a2 b 2 0 c 2. and. a1 a

Problem Set Mash 1 Section 1.2 15. Find a set of generators and relations for Z/nZ. h 1 1 n 0i Z/nZ. Section 1.4 a b 10. Let G 0 c a, b, c 2 R,a6 0,c6 0. a1 b (a) Compute the product of 1 a2 b and 2 0

### Algebra Exam Syllabus

Algebra Exam Syllabus The Algebra comprehensive exam covers four broad areas of algebra: (1) Groups; (2) Rings; (3) Modules; and (4) Linear Algebra. These topics are all covered in the first semester graduate

### Homework #11 Solutions

Homework #11 Solutions p 166, #18 We start by counting the elements in D m and D n, respectively, of order 2. If x D m and x 2 then either x is a flip or x is a rotation of order 2. The subgroup of rotations

### Math Exam 1 Solutions October 12, 2010

Math 415.5 Exam 1 Solutions October 1, 1 As can easily be expected, the solutions provided below are not the only ways to solve these problems, and other solutions may be completely valid. If you have

### Group Theory. 1. Show that Φ maps a conjugacy class of G into a conjugacy class of G.

Group Theory Jan 2012 #6 Prove that if G is a nonabelian group, then G/Z(G) is not cyclic. Aug 2011 #9 (Jan 2010 #5) Prove that any group of order p 2 is an abelian group. Jan 2012 #7 G is nonabelian nite