# MODEL ANSWERS TO HWK #4. ϕ(ab) = [ab] = [a][b]

Size: px
Start display at page:

Download "MODEL ANSWERS TO HWK #4. ϕ(ab) = [ab] = [a][b]"

## Transcription

1 MODEL ANSWERS TO HWK #4 1. (i) Yes. Given a and b Z, ϕ(ab) = [ab] = [a][b] = ϕ(a)ϕ(b). This map is clearly surjective but not injective. Indeed the kernel is easily seen to be nz. (ii) No. Suppose that G is not abelian and that x and y are two elements of G such that xy yx. Then x 1 y 1 y 1 x 1. On the other hand ϕ(xy) = (xy) 1 = y 1 x 1 x 1 y 1 = ϕ(x)ϕ(y), and one wrong certainly does not make a right. (iii) Yes. Suppose that x and y are in G. As G is abelian ϕ(xy) = (xy) 1 = y 1 x 1 = x 1 y 1 = ϕ(x)ϕ(y). Thus ϕ is a homomorphism. ϕ is its own inverse, since (a 1 ) 1 = a and ϕ is a bijection. In particular the kernel of ϕ is trivial. (iv) Yes. ϕ is a homomorphism as the product of two positive numbers is positive, the product of two negative numbers is positive and the product of a negative and a positive number is negative. This map is clearly surjective. The kernel consists of all positive real numbers. Thus ϕ is far from injective. 1

2 (v) Yes. Suppose that x and y are in G. Then ϕ(xy) = (xy) n = x n y n = ϕ(x)ϕ(y). In general this map is neither injective nor surjective. For example, if G = Z and n = 2 then the image of ϕ is 2Z, and for example 1 is not in the image. Now suppose that G = Z 4 and n = 2. Then 2[2] = [4] = [0], so that [2] is in the kernel. In general the kernel of ϕ is Ker ϕ = { g G g n = e }. 2. We need to check that aha 1 = H for all Since H is generated by R, a G = { I, R, R 2, R 3, S 1, S 2, D 1, D 2 }. H = R = { I, R, R 2, R 3 }, it suffices to check that ara 1 H. If we pick a H there is nothing to prove. By symmetry we only need to worry about a = S 1 and a = D 1. S 1 RS 1 1 = S 1 RS 1 = D 1 S 1 = R 3 and D 1 RD 1 1 = D 1 RD 1 = S 2 D 1 = R 3, which is in H. Thus H G. 3. Let g G. We want to show that gzg 1 Z. Pick z Z. Then z commutes with g, so that gzg 1 = zgg 1 = z Z. Thus Z is normal in G. 4. Let g G. We want to show that g(h K)g 1 H K. Pick l H K. Then l H and l K. It follows that glg 1 H and glg 1 K, as both H and K are normal in G. But then glg 1 H K and so H K is normal. 5. H = {e, (1, 2)}. Then the left cosets of H are (i) H = {e, (1, 2)} (1, 3)H = {(1, 3), (1, 2, 3)} (2, 3)H = {(2, 3), (1, 3, 2)} and the right cosets are 2

3 (ii) H = {e, (1, 2)} H(1, 3) = {(1, 3), (1, 3, 2)} H(2, 3) = {(2, 3), (1, 2, 3)}. (iii) Clearly not every left coset is a right coset. For example {(1, 3), (1, 2, 3)} is a left coset, but not a right coset. 6. (i) Suppose that a and b G. Let c = ab and σ = σ a = ψ(a), τ = σ b = ψ(b) and ρ = σ c = ψ(c). We want to check that ρ = στ. Since both sides are permutations of G, it suffices to check that both sides have the same effect on an arbitrary element g G. (σ τ)(g) = σ(τ(g)) = σ(bgb 1 ) = a(bgb 1 )a 1 = (ab)g(ab) 1 = cgc 1 = ρ(g). Thus ψ is a group homomorphism. (ii) Suppose that z Z(G). Let σ = σ z = ψ(z). Then σ(g) = zgz 1 = gzz 1 = g. Thus σ is the identity permutation and z Ker ψ. Thus Z(G) Ker ψ. Now suppose a Ker ψ. Let σ = σ a = ψ(a). Then σ is the identity permutation, so that g = σ(g) = aga 1, for any g G. But then ga = ag for all g G so that a Z(G). Thus Ker ψ Z(G), so that Ker ψ = Z(G). 7. Let g G. We have to show that gθ(n)g 1 θ(n). Now as θ is surjective, we may write g = θ(h), for some h G. Pick m θ(n). Then m = θ(n), for some n N. We have gmg 1 = θ(h)θ(n)θ(h) 1 = θ(hnh 1 ). 3

4 Now hnh 1 N as N is normal. So gmg 1 θ(n) and θ(n) is normal in G. 8. Note that S 3 is the group of permutations of three objects. So we want to find three things on which G acts. Pick any element h of G. Then the order of h divides the order of G. As the order of G is six, it follows that the order of h is one, two, three, or six. It cannot be six, as then G would be cyclic, whence abelian, and it can only be one if h is the identity. We first try to prove that G contains an element of order 2. Suppose not. Let a be an element of G, not the identity. Then H 1 = a = { e, a, a 2 } contains three elements. Pick an element b of G not an element of H 1. Then H 2 = b = { e, b, b 2 } contains three elements, two of which, b and b 2, are not elements of H 1. Thus H 1 H 2 has five elements. The last element c of G must have order two, a contradiction. Thus G contains an element of order 2. Suppose that a has order two. Let H = a = { e, a }, a subgroup of G of order two. Pick an element b which does no belong to H. Consider the group generated by a and b, K = a, b. This has at least three elements, e, a and b. The order of K divides G, so that K has order 3 or 6, by Lagrange. K contains H, so that the order of K is even. It follows that K has order 6, so that G = a, b. As G is not abelian, a and b don t commute, ab ba. The number of left cosets of H in G (the index of H in G) is equal to three, by Lagrange. Let S be the set of left cosets. Define a map from G to A(S) as follows, φ: G A(S) by sending g to σ = φ(g), where σ is the map, σ : S S σ(xh) = gxh, that is, σ acts on the left cosets by left multiplication by g. Suppose that xh = yh, then y = xh and (gy) = (gx)h so that (gx)h = (gy)h and φ is well-defined. σ is a bijection, as its inverse τ is given by left multiplication by g 1. Now we check that φ is a homomorphism. Suppose that g 1 and g 2 are two elements of G. Set σ i = φ(g i ) and let τ = φ(g 1 g 2 ). We need to check that τ = σ 1 σ 2. Pick a left coset xh. Then σ 1 σ 2 (xh) = σ 1 (g 2 xh) = g 1 g 2 xh = τ(xh). 4

5 Thus φ is a homomorphism. We check that φ is injective. It suffices to prove that the kernel of φ is trivial. Pick g Ker φ. Then σ = φ(g) is the identity permutation, so that for every left coset xh, gxh = xh. Consider the left coset H. Then gh = H. It follows that g H, so that either g = e or g = a. If g = a, then consider the left coset bh. We would then have abh = bh, so that ab = bh, where h H. So h = e or h = a. If h = e, then ab = b, and a = e, a contradiction. Otherwise ab = ba, a contradiction. Thus g = e, the kernel of φ is trivial and φ is injective. As both G and A(S) have order six and φ is injective, it follows that φ is a bijection. Hence G is isomorphic to S Let G be a group of order nine. Let g G be an element of G. Then the order of g divides the order of G. Thus the order of g is 1, 3 or 9. If G is cyclic then G is certainly abelian. Thus we may assume that there is no element of order nine. On the other hand the order of g is one iff g = e. Thus we may assume that every element of G, other than the identity, has order three. Let a G be an element of G, other than the identity. Let H = a. Then H has order three. Let S be the set of left cosets of H in G. By Lagrange S has three elements. Let φ: G A(S) S 3 be the map given by left multiplication. As in question 8, φ is a group homomorphism. Let G be the order of the image. Then G divides the order of G, by Lagrange and it also divides the order of S 3. Thus G must have order three. It follows that the kernel of φ has order three. Thus the kernel of φ is H and H is a normal subgroup of G. Let b G be any element of G. Then bab 1 must be an element of H, as H is normal in G. It is clear that bab 1 e. If bab 1 = a then ba = ab, so that a and b commute. If bab 1 = a 2 then b 1 ab = b 2 ab 2 = b(bab 1 )b 1 = ba 2 b 1 = (bab 1 )(bab 1 ) = a, and so ab = ba. Therefore G is abelian. 10. Let S be the set of left cosets of H in G. Define a map φ: G A(S) 5

6 by sending g G to the permutation σ A(S), a map S S defined by the rule σ(ah) = gah. As in question 8, φ is a homomorphism. Let N be the kernel of φ. Then N is normal in G. Suppose that n N and let σ = φ(n). Then σ is the identity permutation of S. In particular σ(h) = H, so that nh = H. Thus n H and so N H. Let n be the index of H, so that the image of G has at most n! elements. In this case there are at most n! left cosets of N in G, since each left coset of N in G is mapped to a different element of A(S). 11. Let A be the set of elements such that φ(a) = a 1. Pick an element g G and let B = g 1 A. Then A B = A + B A B > (3/4) G + (3/4) G G = (1/2) G. Now suppose that g A. If h A B then gh A. It follows that h 1 g 1 = (gh) 1 = φ(gh) = φ(g)φ(h) = g 1 h 1. Taking inverses, we see that g and h must commute. Let C be the centraliser of g. Then A B C, so that C contains more than half the elements of G. On the other hand, C is a subgroup of G. By Lagrange the order of C divides the order of G. Thus C = G. Hence g is in the centre Z of G and so the centre Z of G contains at least 3/4 of the elements of G. But then the centre of G must also equal G, as it is also a subgroup of G. Thus G is abelian. 6

### MODEL ANSWERS TO THE FIFTH HOMEWORK

MODEL ANSWERS TO THE FIFTH HOMEWORK 1. Chapter 3, Section 5: 1 (a) Yes. Given a and b Z, φ(ab) = [ab] = [a][b] = φ(a)φ(b). This map is clearly surjective but not injective. Indeed the kernel is easily

### 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:

### Solutions for Assignment 4 Math 402

Solutions for Assignment 4 Math 402 Page 74, problem 6. Assume that φ : G G is a group homomorphism. Let H = φ(g). We will prove that H is a subgroup of G. Let e and e denote the identity elements of G

### 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

### 2MA105 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

### 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

### 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

### Solution Outlines for Chapter 6

Solution Outlines for Chapter 6 # 1: Find an isomorphism from the group of integers under addition to the group of even integers under addition. Let φ : Z 2Z be defined by x x + x 2x. Then φ(x + y) 2(x

### 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

### 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,

### 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

### MA441: Algebraic Structures I. Lecture 14

MA441: Algebraic Structures I Lecture 14 22 October 2003 1 Review from Lecture 13: We looked at how the dihedral group D 4 can be viewed as 1. the symmetries of a square, 2. a permutation group, and 3.

### 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.

### Mathematics 331 Solutions to Some Review Problems for Exam a = c = 3 2 1

Mathematics 331 Solutions to Some Review Problems for Exam 2 1. Write out all the even permutations in S 3. Solution. The six elements of S 3 are a =, b = 1 3 2 2 1 3 c =, d = 3 2 1 2 3 1 e =, f = 3 1

### 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 9: Group actions

Chapter 9: Group actions Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4120, Summer I 2014 M. Macauley (Clemson) Chapter 9: Group actions

### (5.11) (Second Isomorphism Theorem) If K G and N G, then K/(N K) = NK/N. PF: Verify N HK. Find a homomorphism f : K HK/N with ker(f) = (N K).

Lecture Note of Week 3 6. Normality, Quotients and Homomorphisms (5.7) A subgroup N satisfying any one properties of (5.6) is called a normal subgroup of G. Denote this fact by N G. The homomorphism π

### Algebra homework 6 Homomorphisms, isomorphisms

MATH-UA.343.005 T.A. Louis Guigo Algebra homework 6 Homomorphisms, isomorphisms Exercise 1. Show that the following maps are group homomorphisms and compute their kernels. (a f : (R, (GL 2 (R, given by

### Introduction to Groups

Introduction to Groups Hong-Jian Lai August 2000 1. Basic Concepts and Facts (1.1) A semigroup is an ordered pair (G, ) where G is a nonempty set and is a binary operation on G satisfying: (G1) a (b c)

### Group Theory. Hwan Yup Jung. Department of Mathematics Education, Chungbuk National University

Group Theory Hwan Yup Jung Department of Mathematics Education, Chungbuk National University Hwan Yup Jung (CBNU) Group Theory March 1, 2013 1 / 111 Groups Definition A group is a set G with a binary operation

### 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

### Modern Algebra Homework 9b Chapter 9 Read Complete 9.21, 9.22, 9.23 Proofs

Modern Algebra Homework 9b Chapter 9 Read 9.1-9.3 Complete 9.21, 9.22, 9.23 Proofs Megan Bryant November 20, 2013 First Sylow Theorem If G is a group and p n is the highest power of p dividing G, then

### Homomorphisms. The kernel of the homomorphism ϕ:g G, denoted Ker(ϕ), is the set of elements in G that are mapped to the identity in G.

10. Homomorphisms 1 Homomorphisms Isomorphisms are important in the study of groups because, being bijections, they ensure that the domain and codomain groups are of the same order, and being operation-preserving,

### Your 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.

### Section III.15. Factor-Group Computations and Simple Groups

III.15 Factor-Group Computations 1 Section III.15. Factor-Group Computations and Simple Groups Note. In this section, we try to extract information about a group G by considering properties of the factor

### Solutions to Assignment 4

1. Let G be a finite, abelian group written additively. Let x = g G g, and let G 2 be the subgroup of G defined by G 2 = {g G 2g = 0}. (a) Show that x = g G 2 g. (b) Show that x = 0 if G 2 = 2. If G 2

### Answers to Final Exam

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

### Theorems and Definitions in Group Theory

Theorems and Definitions in Group Theory Shunan Zhao Contents 1 Basics of a group 3 1.1 Basic Properties of Groups.......................... 3 1.2 Properties of Inverses............................. 3

### 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

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 =

### Normal Subgroups and Factor Groups

Normal Subgroups and Factor Groups Subject: Mathematics Course Developer: Harshdeep Singh Department/ College: Assistant Professor, Department of Mathematics, Sri Venkateswara College, University of Delhi

### Part II Permutations, Cosets and Direct Product

Part II Permutations, Cosets and Direct Product Satya Mandal University of Kansas, Lawrence KS 66045 USA January 22 8 Permutations Definition 8.1. Let A be a set. 1. A a permuation of A is defined to be

### Lecture 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

### Lecture 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

### 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

### Zachary Scherr Math 370 HW 7 Solutions

1 Book Problems 1. 2.7.4b Solution: Let U 1 {u 1 u U} and let S U U 1. Then (U) is the set of all elements of G which are finite products of elements of S. We are told that for any u U and g G we have

### 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

### Automorphism Groups Definition. An automorphism of a group G is an isomorphism G G. The set of automorphisms of G is denoted Aut G.

Automorphism Groups 9-9-2012 Definition. An automorphism of a group G is an isomorphism G G. The set of automorphisms of G is denoted Aut G. Example. The identity map id : G G is an automorphism. Example.

### MATH 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

### 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

### Normal Subgroups and Quotient Groups

Normal Subgroups and Quotient Groups 3-20-2014 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

### MA441: 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

### Isomorphisms. 0 a 1, 1 a 3, 2 a 9, 3 a 7

Isomorphisms Consider the following Cayley tables for the groups Z 4, U(), R (= the group of symmetries of a nonsquare rhombus, consisting of four elements: the two rotations about the center, R 8, and

### DISCRETE MATH (A LITTLE) & BASIC GROUP THEORY - PART 3/3. Contents

DISCRETE MATH (A LITTLE) & BASIC GROUP THEORY - PART 3/3 T.K.SUBRAHMONIAN MOOTHATHU Contents 1. Cayley s Theorem 1 2. The permutation group S n 2 3. Center of a group, and centralizers 4 4. Group actions

### SUPPLEMENT 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

### First Semester Abstract Algebra for Undergraduates

First Semester Abstract Algebra for Undergraduates Lecture notes by: Khim R Shrestha, Ph. D. Assistant Professor of Mathematics University of Great Falls Great Falls, Montana Contents 1 Introduction to

### 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 =

### Solutions to odd-numbered exercises Peter J. Cameron, Introduction to Algebra, Chapter 3

Solutions to odd-numbered exercises Peter J. Cameron, Introduction to Algebra, Chapter 3 3. (a) Yes; (b) No; (c) No; (d) No; (e) Yes; (f) Yes; (g) Yes; (h) No; (i) Yes. Comments: (a) is the additive group

### 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

### Math 250A, Fall 2004 Problems due October 5, 2004 The problems this week were from Lang s Algebra, Chapter I.

Math 250A, Fall 2004 Problems due October 5, 2004 The problems this week were from Lang s Algebra, Chapter I. 24. We basically know already that groups of order p 2 are abelian. Indeed, p-groups have non-trivial

### 1.1 Definition. A monoid is a set M together with a map. 1.3 Definition. A monoid is commutative if x y = y x for all x, y M.

1 Monoids and groups 1.1 Definition. A monoid is a set M together with a map M M M, (x, y) x y such that (i) (x y) z = x (y z) x, y, z M (associativity); (ii) e M such that x e = e x = x for all x M (e

### Math 3140 Fall 2012 Assignment #4

Math 3140 Fall 2012 Assignment #4 Due Fri., Sept. 28. Remember to cite your sources, including the people you talk to. In this problem set, we use the notation gcd {a, b} for the greatest common divisor

### Quiz 2 Practice Problems

Quiz 2 Practice Problems Math 332, Spring 2010 Isomorphisms and Automorphisms 1. Let C be the group of complex numbers under the operation of addition, and define a function ϕ: C C by ϕ(a + bi) = a bi.

### Groups Subgroups Normal subgroups Quotient groups Homomorphisms Cyclic groups Permutation groups Cayley s theorem Class equations Sylow theorems

Group Theory Groups Subgroups Normal subgroups Quotient groups Homomorphisms Cyclic groups Permutation groups Cayley s theorem Class equations Sylow theorems Groups Definition : A non-empty set ( G,*)

### Exercises 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

### Group Theory

Group Theory 2014 2015 Solutions to the exam of 4 November 2014 13 November 2014 Question 1 (a) For every number n in the set {1, 2,..., 2013} there is exactly one transposition (n n + 1) in σ, so σ is

### Math 103 HW 9 Solutions to Selected Problems

Math 103 HW 9 Solutions to Selected Problems 4. Show that U(8) is not isomorphic to U(10). Solution: Unfortunately, the two groups have the same order: the elements are U(n) are just the coprime elements

### 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

### CHAPTER 9. Normal Subgroups and Factor Groups. Normal Subgroups

Normal Subgroups CHAPTER 9 Normal Subgroups and Factor Groups If H apple G, we have seen situations where ah 6= Ha 8 a 2 G. Definition (Normal Subgroup). A subgroup H of a group G is a normal subgroup

### The Gordon game. EWU Digital Commons. Eastern Washington University. Anthony Frenk Eastern Washington University

Eastern Washington University EWU Digital Commons EWU Masters Thesis Collection Student Research and Creative Works 2013 The Gordon game Anthony Frenk Eastern Washington University Follow this and additional

### Lecture 7 Cyclic groups and subgroups

Lecture 7 Cyclic groups and subgroups Review Types of groups we know Numbers: Z, Q, R, C, Q, R, C Matrices: (M n (F ), +), GL n (F ), where F = Q, R, or C. Modular groups: Z/nZ and (Z/nZ) Dihedral groups:

### Math 581 Problem Set 9

Math 581 Prolem Set 9 1. Let m and n e relatively prime positive integers. (a) Prove that Z/mnZ = Z/mZ Z/nZ as RINGS. (Hint: First Isomorphism Theorem) Proof: Define ϕz Z/mZ Z/nZ y ϕ(x) = ([x] m, [x] n

### Groups. Chapter 1. If ab = ba for all a, b G we call the group commutative.

Chapter 1 Groups A group G is a set of objects { a, b, c, } (not necessarily countable) together with a binary operation which associates with any ordered pair of elements a, b in G a third element ab

### Master Algèbre géométrie et théorie des nombres Final exam of differential geometry Lecture notes allowed

Université de Bordeaux U.F. Mathématiques et Interactions Master Algèbre géométrie et théorie des nombres Final exam of differential geometry 2018-2019 Lecture notes allowed Exercise 1 We call H (like

### The number 6. Gabriel Coutinho 2013

The number 6 Gabriel Coutinho 2013 Abstract The number 6 has a unique and distinguished property. It is the only natural number n for which there is a construction of n isomorphic objects on a set with

### Math 120 HW 9 Solutions

Math 120 HW 9 Solutions June 8, 2018 Question 1 Write down a ring homomorphism (no proof required) f from R = Z[ 11] = {a + b 11 a, b Z} to S = Z/35Z. The main difficulty is to find an element x Z/35Z

### 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

### MAT534 Fall 2013 Practice Midterm I The actual midterm will consist of five problems.

MAT534 Fall 2013 Practice Midterm I The actual midterm will consist of five problems. Problem 1 Find all homomorphisms a) Z 6 Z 6 ; b) Z 6 Z 18 ; c) Z 18 Z 6 ; d) Z 12 Z 15 ; e) Z 6 Z 25 Proof. a)ψ(1)

### MA441: Algebraic Structures I. Lecture 15

MA441: Algebraic Structures I Lecture 15 27 October 2003 1 Correction for Lecture 14: I should have used multiplication on the right for Cayley s theorem. Theorem 6.1: Cayley s Theorem Every group is isomorphic

### Course 311: Abstract Algebra Academic year

Course 311: Abstract Algebra Academic year 2007-08 D. R. Wilkins Copyright c David R. Wilkins 1997 2007 Contents 1 Topics in Group Theory 1 1.1 Groups............................... 1 1.2 Examples of Groups.......................

### MATH 3005 ABSTRACT ALGEBRA I FINAL SOLUTION

MATH 3005 ABSTRACT ALGEBRA I FINAL SOLUTION SPRING 2014 - MOON Write your answer neatly and show steps. Any electronic devices including calculators, cell phones are not allowed. (1) Write the definition.

### MATH 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.

### 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

### Selected exercises from Abstract Algebra by Dummit and Foote (3rd edition).

Selected exercises from Abstract Algebra by Dummit Foote (3rd edition). Bryan Félix Abril 12, 2017 Section 4.1 Exercise 1. Let G act on the set A. Prove that if a, b A b = ga for some g G, then G b = gg

### Assigment 1. 1 a b. 0 1 c A B = (A B) (B A). 3. In each case, determine whether G is a group with the given operation.

1. Show that the set G = multiplication. Assigment 1 1 a b 0 1 c a, b, c R 0 0 1 is a group under matrix 2. Let U be a set and G = {A A U}. Show that G ia an abelian group under the operation defined by

### CONSEQUENCES 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.

### A 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

### Algebra-I, Fall Solutions to Midterm #1

Algebra-I, Fall 2018. Solutions to Midterm #1 1. Let G be a group, H, K subgroups of G and a, b G. (a) (6 pts) Suppose that ah = bk. Prove that H = K. Solution: (a) Multiplying both sides by b 1 on the

### 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 210A: Algebra, Homework 6

Math 210A: Algebra, Homework 6 Ian Coley November 13, 2013 Problem 1 For every two nonzero integers n and m construct an exact sequence For which n and m is the sequence split? 0 Z/nZ Z/mnZ Z/mZ 0 Let

### Two subgroups and semi-direct products

Two subgroups and semi-direct products 1 First remarks Throughout, we shall keep the following notation: G is a group, written multiplicatively, and H and K are two subgroups of G. We define the subset

### Algebra I Notes. Clayton J. Lungstrum. July 18, Based on the textbook Algebra by Serge Lang

Algebra I Notes Based on the textbook Algebra by Serge Lang Clayton J. Lungstrum July 18, 2013 Contents Contents 1 1 Group Theory 2 1.1 Basic Definitions and Examples......................... 2 1.2 Subgroups.....................................

### Section 13 Homomorphisms

Section 13 Homomorphisms Instructor: Yifan Yang Fall 2006 Homomorphisms Definition A map φ of a group G into a group G is a homomorphism if for all a, b G. φ(ab) = φ(a)φ(b) Examples 1. Let φ : G G be defined

### 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

### SUMMARY ALGEBRA I LOUIS-PHILIPPE THIBAULT

SUMMARY ALGEBRA I LOUIS-PHILIPPE THIBAULT Contents 1. Group Theory 1 1.1. Basic Notions 1 1.2. Isomorphism Theorems 2 1.3. Jordan- Holder Theorem 2 1.4. Symmetric Group 3 1.5. Group action on Sets 3 1.6.

### Frank Moore Algebra 901 Notes Professor: Tom Marley Direct Products of Groups:

Frank Moore Algebra 901 Notes Professor: Tom Marley Direct Products of Groups: Definition: The external direct product is defined to be the following: Let H 1,..., H n be groups. H 1 H 2 H n := {(h 1,...,

### Group Theory and Applications MST30010

Group Theory and Applications MST30010 ii Contents 1 Material from last year 1 2 Normal subgroups and quotient groups 3 3 Morphims 7 4 Subgroups of quotient groups 13 5 Group actions 15 5.1 Size of orbits...........................

### Cosets 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

### Author: Bob Howlett Group representation theory Lecture 1, 28/7/97. Introduction

Author: Bob Howlett Group representation theory Lecture 1, 28/7/97 Introduction This course is a mix of group theory and linear algebra, with probably more of the latter than the former. You may need to

### A SIMPLE PROOF OF BURNSIDE S CRITERION FOR ALL GROUPS OF ORDER n TO BE CYCLIC

A SIMPLE PROOF OF BURNSIDE S CRITERION FOR ALL GROUPS OF ORDER n TO BE CYCLIC SIDDHI PATHAK Abstract. This note gives a simple proof of a famous theorem of Burnside, namely, all groups of order n are cyclic

### 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 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

### CS 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

### 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

### Examples: The (left or right) cosets of the subgroup H = 11 in U(30) = {1, 7, 11, 13, 17, 19, 23, 29} are

Cosets Let H be a subset of the group G. (Usually, H is chosen to be a subgroup of G.) If a G, then we denote by ah the subset {ah h H}, the left coset of H containing a. Similarly, Ha = {ha h H} is the

### AM 106/206: Applied Algebra Madhu Sudan 1. Lecture Notes 11

AM 106/206: Applied Algebra Madhu Sudan 1 Lecture Notes 11 October 17, 2016 Reading: Gallian Chapters 9 & 10 1 Normal Subgroups Motivation: Recall that the cosets of nz in Z (a+nz) are the same as the

### To hand in: (a) Prove that a group G is abelian (= commutative) if and only if (xy) 2 = x 2 y 2 for all x, y G.

Homework #6. Due Thursday, October 14th Reading: For this homework assignment: Sections 3.3 and 3.4 (up to page 167) Before the class next Thursday: Sections 3.5 and 3.4 (pp. 168-171). Also review the

### M3P10: GROUP THEORY LECTURES BY DR. JOHN BRITNELL; NOTES BY ALEKSANDER HORAWA

M3P10: GROUP THEORY LECTURES BY DR. JOHN BRITNELL; NOTES BY ALEKSANDER HORAWA These are notes from the course M3P10: Group Theory taught by Dr. John Britnell, in Fall 2015 at Imperial College London. They

### Chapter 5 Groups of permutations (bijections) Basic notation and ideas We study the most general type of groups - groups of permutations

Chapter 5 Groups of permutations (bijections) Basic notation and ideas We study the most general type of groups - groups of permutations (bijections). Definition A bijection from a set A to itself is also