# (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).

Size: px
Start display at page:

Download "(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)."

Transcription

1 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 in the proof of (v) = (vi) in (5.6) is called the natural projection or canonical homomorphism of G onto G/N. (5.8) Let φ : G H be a homomorphism. (i) if H H, then φ 1 (H ) G. (ii) If G G, then φ(g ) H. Proof (i) Use (2.3) and (ii) of (5.1). (ii) of (5.8) = (v) of (5.1). (5.9) For any H G/N, N π 1 ( H) G. Proof It suffices to show that π 1 ( H) G. Use (i) of (5.8). (5.10) (Thm 5.6) Let f : G H be a group homomorphism and N G such that N < ker(f). Then there exists a unique homomorphism f : G/N H such that (i) f(an) = f(a), a G. (ii) Im(f) = Im( f) and ker( f) = ker(f)/n. Moreover, (First Isomorphism Theorem) f is an isomorphism iff f is an epimorphism and N = ker(f). (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). (5.12) (Third Isomorphism Theorem) If K G and N G and if K < H then H/K G/K and (G/K)/(H/K) = G/H. PF: Verify H/K G/K. Find a homomorphism f : G/K G/H with ker(f) = H/K. (5.13) (Thm 5.11) Let f : G H be an epimorphism (onto homomorphism) of groups. Then the assignment K f(k) is (i) a bijection between the set S f (G) = {K G : ker(f) K G} and the set S(H) = {N H}; and (ii) a bijection between the set S f (G) = {K G : ker(f) K G} and the set S(H) = {N H}. 1

2 (5.14) Z(G) G. (5.14a) If G/Z(G) is cyclic, then G is abelian. (5.15) A group G is simple if G > 1 and if H {< 1 >, G} whenever H G. (5.15a) The only simple abelian groups are Z p, for prime p s. Proof (5.1). (5.16) An example: being normal is not a transitive relation. Let G = D 8 =< r, s r 4 = 1, s 2 = 1, rs = sr 1 >. Let H = {1, r 2, s, sr 2 }, and K = {1, s}. Then since G : H = 2, and H : K = 2, both K H and H G. However, rsr 1 = r 2 s K and so K G. 6. Symmetric, Alternating, and Dihedral Groups (6.1) Review of symmetric group on n elements. Any permutation φ S n is a product of cycles (called the cycle decomposition of φ). A 2-cycle is a transposition. Any permutation can be written as a product of transpositions. (i 1 i 2 i k ) = (i 1 i k )(i 1 i k 1 ) (i 1 i 3 )(i 1 i 2 ). (6.2) For each n 3, the dihedral group D n is a subgroup permutations of order 2n generated by Any group with two generator a, b with the relations a n = e, b 2 = e and ba = a 1 b is isomorphic to D n, and is also called the dihedral group of order 2n. Example: D 4. (6.3) No φ S n (n 2) can be expressed both as a product of an even number of transpositions and as a product of an odd number of transpositions. Proof We use e to denote the identity of S n. (Step 1) (6.2) holds for φ = e. Suppose that S n is the set of all permutations on the set {1, 2,, n}, and that e = τ k τ 1, where each τ i is a transposition. Let X = {x : 1 x n and x is involved in some of the τ s }, and s = X. 2

3 Argue by induction on s. If s = 2, then we may assume that the involved letters are 1 and 2, and e = τ k τ 1, where each τ i = (1, 2). Since e = (1, 2)(1, 2), k must be even. Assume that s 3 and that (Step 1) holds for smaller values of s. Suppose that e = τ k τ 1, where each τ i is a transposition, and where the involved letters are in {1, 2,..., s}. We further argue by induction on k. (Step 1) holds trivially if k = 2, and so we assume further that (Step 1) holds for smaller values of k. Pick m X. Let τ j be the 1st transposition (from R to L) that contains m. Then τ j+1 τ j must be one in the left side of (x, m)(x, m) = e (m, y)(m, x) = (m, x)(x, y) (y, z)(m, x) = (m, x)(y, z) (x, y)(x, m) = (m, y)(x, y) Hence the substitution of the left by the right either reduces the number of transpositions by 2; whence by induction on k, (Step 1) holds; or moves the 1st transposition containing m to the left by one step. Repeat this process (assuming that k remains unchanged) until the first τ j containing m is τ k 1. Then τ t au k 1 must be one of the four cases listed above. In this case, only the case τ k = τ k 1 = (x, m) will occur, as otherwise, after the process of pushing m to the left, the right most transposition of a factoring of e is the only transposition in the factorization of e contains the element m, and so m must be moved, contrary to the fact that e S n is the identity permutation. Therefore, such a process can eliminate the element m, without introducing any new elements involved in the factorization, and without changing the parity of k. becomes X 1, and so by induction on X, (Step 1) holds also for all values of k. (Step 2) General Case: Suppose φ S n has two factorizations: φ = τ 1 τ 2 τ r = τ 1τ 2 τ t, Now X where τ i s and τ j s are transpositions. Then φ 1 = τr 1 τ1 1 and so e = φφ 1 = τr 1 τ1 1 τ 1 τ 2 τ t. Hence r + t must be even, and so r and t must have the same parity. (6.4) (Even and Odd Permutations) A permutation in S n is even (or odd) if it can be expressed as a product of an even (or odd) number of transpositions. The set of all even permutations in S n is denoted by A n. A n is a subgroup of S n, called the Alternating Group of degree n. Proof Use (2.3) to show A n S n. 3

4 (6.5) Let a = (123 n) and (1n)(2(n 1)) ( n 2 b = ( n 2 + 1)) (1n)(2(n 1)) ( n 1 n if n is even, n+1 )( 2 ) if n is odd. The subgroup D 2n =< a, b > is called the dihedral group of order 2n. The presentation of D 2n is a n = b 2 = 1, and ba = a 1 b. We denote that D 2n =< a, b a n = b 2 = 1, and ba = a 1 b >. 8. Direct Products and Direct Sums (8.1) The direct product (also refereed as complete direct sum) of a collection of groups G i, i I consists of the Cartesian product (of sets) G i = {f : I i I G i such that for each i I, f(i) G i } i I in which the binary operation is defined componentwise. That is, if f, g i I G i, then for each i I, fg(i) = f(i)g(i) with the multiplication taking place in G i. One can routinely verify that this is a group. Elements in i I G i are also commonly written in a vector form a = {a i }, where a i = a(i). In this case, the product (sum, if in additive notation) of {a i } and {b i } will be {a i b i } (or respectively, {a i + b i }, in additive notation.) (8.2) For a fixed i I, let J = I {i}. Define a map ι i : G i j I G j by defining ι i (g) to be the element in j I G j such that for any j I, g ι i (g)(j) = e Gj if j = i otherwise Then ι i is an injective homomorphism, (referred as the inclusion map, or the canonical injection), which is an isomorphism between G i and a subgroup ι i (G i ). (8.2A) Let φ i : j I G j j J G j by φ(f)(j) = f(j) if j i and φ(f)(j) = e Gi if j = i. Then φ i is an onto homomorphism with kernel ι i (G i ). Therefore, j I G j/ι i (G i ) = j J G j. (8.3) Fix i I. Define π i : j J G j G i by π i (f)(j) = f(i) if j = i and π i (f)(j) = e Gj if 4

5 j i. Then π i is also a homomorphism, (referred as the canonical projection map onto the ith component). (8.4) Let w G i = {f G i : for all but finitely many i I, f(i) = eg i }. i I i I Then w i I G i is also a group, called the (external) weak direct product (also referred as the (external) direct sum) of the G i s. (8.5) Let {N i i I} be a collection of normal subgroups of a group G. If each of the following holds: (i) G = i I N i. (ii) for each k I, N k i I {k} N i = {e G }. Then G = w i I N i. Proof For each f w i I N i, there is a finite set I f I such that if j I I f, then f(j) = e Nj. Define a map φ : w i I N i G by φ(f) = i I f f(i). As I f <, φ(f) is a well-defined product in G. By (ii), when i j, a i N i and a j N j, a i a j = a j a i. This, together with (i), implies that φ is an onto homomorphism. Note Ker φ = {f w i I N i i I f f(i) = e G }. Apply (ii) again to see that Ker φ = {e}, where e(i) = e Ni, i I. (8.6) Let G and the N i s satisfy the hypotheses (i) and (ii) in (1.4). Then G is called the internal weak direct product (also referred as the internal direct sum) of the N i s. (8.7) Let {f i : G i H i } be a collection of group homomorphisms, and let f = i I f i denote the map f : i I G i i I H i such that for every {a i } i I G i, f({a i }) = {f i (a i )} i I H i. Then f is a homomorphism. such that f( w i I G i w i I H i, Ker f = i I Ker f i, and Im f = i I Im f i. Proof Routine verification. (8.8) Let {G i i I} and {N i i I} be collections of groups such that for each i I, N i G i. Then each of the following holds. (i) i I N i i I G i, and i I G i/ i I N i = i I G i/n i. 5

6 ii w i I N i w i I G i, and w i I G i / w i I N i = wi I G i /N i. Proof Let π i : G i G i /N i be canonical epimorphism. Then i I π i : i I G i i I G i/n i is also an epimorphism with kernel i I N i. Then apply the 1st isomorphism theorem. 6

### 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)

### II. Products of Groups

II. Products of Groups Hong-Jian Lai October 2002 1. Direct Products (1.1) The direct product (also refereed as complete direct sum) of a collection of groups G i, i I consists of the Cartesian product

### 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 =

### 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

### Lecture Note of Week 2

Lecture Note of Week 2 2. Homomorphisms and Subgroups (2.1) Let G and H be groups. A map f : G H is a homomorphism if for all x, y G, f(xy) = f(x)f(y). f is an isomorphism if it is bijective. If f : G

### Algebra. Travis Dirle. December 4, 2016

Abstract Algebra 2 Algebra Travis Dirle December 4, 2016 2 Contents 1 Groups 1 1.1 Semigroups, Monoids and Groups................ 1 1.2 Homomorphisms and Subgroups................. 2 1.3 Cyclic Groups...........................

### 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

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

### Modern Algebra (MA 521) Synopsis of lectures July-Nov 2015 semester, IIT Guwahati

Modern Algebra (MA 521) Synopsis of lectures July-Nov 2015 semester, IIT Guwahati Shyamashree Upadhyay Contents 1 Lecture 1 4 1.1 Properties of Integers....................... 4 1.2 Sets, relations and

### 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

### 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

### 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

### Homework Problems, Math 200, Fall 2011 (Robert Boltje)

Homework Problems, Math 200, Fall 2011 (Robert Boltje) Due Friday, September 30: ( ) 0 a 1. Let S be the set of all matrices with entries a, b Z. Show 0 b that S is a semigroup under matrix multiplication

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

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

### 120A LECTURE OUTLINES

120A LECTURE OUTLINES RUI WANG CONTENTS 1. Lecture 1. Introduction 1 2 1.1. An algebraic object to study 2 1.2. Group 2 1.3. Isomorphic binary operations 2 2. Lecture 2. Introduction 2 3 2.1. The multiplication

### BASIC 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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:

### 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 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

### 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.......................

### S11MTH 3175 Group Theory (Prof.Todorov) Quiz 6 (PracticeSolutions) Name: 1. Let G and H be two groups and G H the external direct product of G and H.

Some of the problems are very easy, some are harder. 1. Let G and H be two groups and G H the external direct product of G and H. (a) Prove that the map f : G H H G defined as f(g, h) = (h, g) is a group

### ALGEBRA QUALIFYING EXAM PROBLEMS

ALGEBRA QUALIFYING EXAM PROBLEMS Kent State University Department of Mathematical Sciences Compiled and Maintained by Donald L. White Version: August 29, 2017 CONTENTS LINEAR ALGEBRA AND MODULES General

### 1.5 Applications Of The Sylow Theorems

14 CHAPTER1. GROUP THEORY 8. The Sylow theorems are about subgroups whose order is a power of a prime p. Here is a result about subgroups of index p. Let H be a subgroup of the finite group G, and assume

### 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

### 2) e = e G G such that if a G 0 =0 G G such that if a G e a = a e = a. 0 +a = a+0 = a.

Chapter 2 Groups Groups are the central objects of algebra. In later chapters we will define rings and modules and see that they are special cases of groups. Also ring homomorphisms and module homomorphisms

### 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

### MATH 436 Notes: Homomorphisms.

MATH 436 Notes: Homomorphisms. Jonathan Pakianathan September 23, 2003 1 Homomorphisms Definition 1.1. Given monoids M 1 and M 2, we say that f : M 1 M 2 is a homomorphism if (A) f(ab) = f(a)f(b) for all

### Abstract Algebra II Groups ( )

Abstract Algebra II Groups ( ) Melchior Grützmann / melchiorgfreehostingcom/algebra October 15, 2012 Outline Group homomorphisms Free groups, free products, and presentations Free products ( ) Definition

### 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.

### 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

### 23.1. Proof of the fundamental theorem of homomorphisms (FTH). We start by recalling the statement of FTH introduced last time.

23. Quotient groups II 23.1. Proof of the fundamental theorem of homomorphisms (FTH). We start by recalling the statement of FTH introduced last time. Theorem (FTH). Let G, H be groups and ϕ : G H a homomorphism.

### 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

### 0 Sets and Induction. Sets

0 Sets and Induction Sets A set is an unordered collection of objects, called elements or members of the set. A set is said to contain its elements. We write a A to denote that a is an element of the set

### MTH 411 Lecture Notes Based on Hungerford, Abstract Algebra

MTH 411 Lecture Notes Based on Hungerford, Abstract Algebra Ulrich Meierfrankenfeld Department of Mathematics Michigan State University East Lansing MI 48824 meier@math.msu.edu August 28, 2014 2 Contents

### 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

### Chapter I: Groups. 1 Semigroups and Monoids

Chapter I: Groups 1 Semigroups and Monoids 1.1 Definition Let S be a set. (a) A binary operation on S is a map b : S S S. Usually, b(x, y) is abbreviated by xy, x y, x y, x y, x y, x + y, etc. (b) Let

### 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

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

### CHAPTER III NORMAL SERIES

CHAPTER III NORMAL SERIES 1. Normal Series A group is called simple if it has no nontrivial, proper, normal subgroups. The only abelian simple groups are cyclic groups of prime order, but some authors

### 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

### 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

### 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

### Cover Page. The handle holds various files of this Leiden University dissertation

Cover Page The handle http://hdl.handle.net/1887/54851 holds various files of this Leiden University dissertation Author: Stanojkovski, M. Title: Intense automorphisms of finite groups Issue Date: 2017-09-05

### 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,

### School of Mathematics and Statistics. MT5824 Topics in Groups. Problem Sheet I: Revision and Re-Activation

MRQ 2009 School of Mathematics and Statistics MT5824 Topics in Groups Problem Sheet I: Revision and Re-Activation 1. Let H and K be subgroups of a group G. Define HK = {hk h H, k K }. (a) Show that HK

### its image and kernel. A subgroup of a group G is a non-empty subset K of G such that k 1 k 1

10 Chapter 1 Groups 1.1 Isomorphism theorems Throughout the chapter, we ll be studying the category of groups. Let G, H be groups. Recall that a homomorphism f : G H means a function such that f(g 1 g

### 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

### SUMMARY OF GROUPS AND RINGS GROUPS AND RINGS III Week 1 Lecture 1 Tuesday 3 March.

SUMMARY OF GROUPS AND RINGS GROUPS AND RINGS III 2009 Week 1 Lecture 1 Tuesday 3 March. 1. Introduction (Background from Algebra II) 1.1. Groups and Subgroups. Definition 1.1. A binary operation on a set

### Problem 1. Let I and J be ideals in a ring commutative ring R with 1 R. Recall

I. Take-Home Portion: Math 350 Final Exam Due by 5:00pm on Tues. 5/12/15 No resources/devices other than our class textbook and class notes/handouts may be used. You must work alone. Choose any 5 problems

### 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

### 23.1. Proof of the fundamental theorem of homomorphisms (FTH). We start by recalling the statement of FTH introduced last time.

23. Quotient groups II 23.1. Proof of the fundamental theorem of homomorphisms (FTH). We start by recalling the statement of FTH introduced last time. Theorem (FTH). Let G, Q be groups and ϕ : G Q a homomorphism.

### Exercises on chapter 1

Exercises on chapter 1 1. Let G be a group and H and K be subgroups. Let HK = {hk h H, k K}. (i) Prove that HK is a subgroup of G if and only if HK = KH. (ii) If either H or K is a normal subgroup of G

### 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.

### 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.

### Algebra. Jim Coykendall

Algebra Jim Coykendall March 2, 2011 2 Chapter 1 Basic Basics The notation N, Z, Q, R, C refer to the natural numbers, the integers, the rational numbers, the real numbers and the complex numbers respectively.

### 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

### 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

### 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

### Cosets, factor groups, direct products, homomorphisms, isomorphisms

Cosets, factor groups, direct products, homomorphisms, isomorphisms Sergei Silvestrov Spring term 2011, Lecture 11 Contents of the lecture Cosets and the theorem of Lagrange. Direct products and finitely

### 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

### 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

### S10MTH 3175 Group Theory (Prof.Todorov) Quiz 6 (Practice) Name: Some of the problems are very easy, some are harder.

Some of the problems are very easy, some are harder. 1. Let F : Z Z be a function defined as F (x) = 10x. (a) Prove that F is a group homomorphism. (b) Find Ker(F ) Solution: Ker(F ) = {0}. Proof: Let

### 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

### 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.

### 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

### 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

### 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

### 2.3 The Krull-Schmidt Theorem

2.3. THE KRULL-SCHMIDT THEOREM 41 2.3 The Krull-Schmidt Theorem Every finitely generated abelian group is a direct sum of finitely many indecomposable abelian groups Z and Z p n. We will study a large

### 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 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

### Joseph Muscat Universal Algebras. 1 March 2013

Joseph Muscat 2015 1 Universal Algebras 1 Operations joseph.muscat@um.edu.mt 1 March 2013 A universal algebra is a set X with some operations : X n X and relations 1 X m. For example, there may be specific

### 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

### 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

### 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

### 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

### Tree-adjoined spaces and the Hawaiian earring

Tree-adjoined spaces and the Hawaiian earring W. Hojka (TU Wien) Workshop on Fractals and Tilings 2009 July 6-10, 2009, Strobl (Austria) W. Hojka (TU Wien) () Tree-adjoined spaces and the Hawaiian earring

### Name: Solutions - AI FINAL EXAM

1 2 3 4 5 6 7 8 9 10 11 12 13 total Name: Solutions - AI FINAL EXAM The first 7 problems will each count 10 points. The best 3 of # 8-13 will count 10 points each. Total is 100 points. A 4th problem from

### 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

### Math 451, 01, Exam #2 Answer Key

Math 451, 01, Exam #2 Answer Key 1. (25 points): If the statement is always true, circle True and prove it. If the statement is never true, circle False and prove that it can never be true. If the statement

GENERATING SETS KEITH CONRAD 1 Introduction In R n, every vector can be written as a unique linear combination of the standard basis e 1,, e n A notion weaker than a basis is a spanning set: a set of vectors

### 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

### 20 Group Homomorphisms

20 Group Homomorphisms In Example 1810(d), we have observed that the groups S 4 /V 4 and S 3 have almost the same multiplication table They have the same structure In this paragraph, we study groups with

### Math 121 Homework 5: Notes on Selected Problems

Math 121 Homework 5: Notes on Selected Problems 12.1.2. Let M be a module over the integral domain R. (a) Assume that M has rank n and that x 1,..., x n is any maximal set of linearly independent elements

### Lectures - XXIII and XXIV Coproducts and Pushouts

Lectures - XXIII and XXIV Coproducts and Pushouts We now discuss further categorical constructions that are essential for the formulation of the Seifert Van Kampen theorem. We first discuss the notion

### 3.2 Modules of Fractions

3.2 Modules of Fractions Let A be a ring, S a multiplicatively closed subset of A, and M an A-module. Define a relation on M S = { (m, s) m M, s S } by, for m,m M, s,s S, 556 (m,s) (m,s ) iff ( t S) t(sm

### D-MATH Algebra I HS 2013 Prof. Brent Doran. Solution 3. Modular arithmetic, quotients, product groups

D-MATH Algebra I HS 2013 Prof. Brent Doran Solution 3 Modular arithmetic, quotients, product groups 1. Show that the functions f = 1/x, g = (x 1)/x generate a group of functions, the law of composition

### TCC Homological Algebra: Assignment #3 (Solutions)

TCC Homological Algebra: Assignment #3 (Solutions) David Loeffler, d.a.loeffler@warwick.ac.uk 30th November 2016 This is the third of 4 problem sheets. Solutions should be submitted to me (via any appropriate

### Problem 1.1. Classify all groups of order 385 up to isomorphism.

Math 504: Modern Algebra, Fall Quarter 2017 Jarod Alper Midterm Solutions Problem 1.1. Classify all groups of order 385 up to isomorphism. Solution: Let G be a group of order 385. Factor 385 as 385 = 5

### 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

### Physics 251 Solution Set 1 Spring 2017

Physics 5 Solution Set Spring 07. Consider the set R consisting of pairs of real numbers. For (x,y) R, define scalar multiplication by: c(x,y) (cx,cy) for any real number c, and define vector addition

### School of Mathematics and Statistics MT5824 Topics in Groups Problem Sheet IV: Composition series and the Jordan Hölder Theorem (Solutions)

CMRD 2010 School of Mathematics and Statistics MT5824 Topics in Groups Problem Sheet IV: Composition series and the Jordan Hölder Theorem (Solutions) 1. Let G be a group and N be a normal subgroup of G.

### S11MTH 3175 Group Theory (Prof.Todorov) Final (Practice Some Solutions) 2 BASIC PROPERTIES

S11MTH 3175 Group Theory (Prof.Todorov) Final (Practice Some Solutions) 2 BASIC PROPERTIES 1 Some Definitions For your convenience, we recall some of the definitions: A group G is called simple if it has