# MATH 436 Notes: Cyclic groups and Invariant Subgroups.

Size: px
Start display at page:

Transcription

1 MATH 436 Notes: Cyclic groups and Invariant Subgroups. Jonathan Pakianathan September 30, Cyclic Groups Now that we have enough basic tools, let us go back and study the structure of cyclic groups. Recall, these are exactly the groups G which can be generated by a single element x G. Before we begin we record an important example: Example 1.1. Given a group G and an element x G we may define a homomorphism φ x : Z G via φ x (n) = x n. It is easy to see that Im(φ x ) =< x > is the cyclic subgroup generated by x. If one considers ker(φ x ), we have seen that it must be of the form dz for some unique integer d 0. We define the order of x, denote o(x) by: { d if d 1 o(x) = if d = 0 Thus if x has infinite order then the elements in the set {x k k Z} are all distinct while if x has order o(x) 1 then x k = e exactly when k is a multiple of o(x). Thus x m = x n whenever m n modulo o(x). By the First Isomorphism Theorem, < x > = Z/o(x)Z. Hence when x has finite order, < x > = o(x) and so in the case of a finite group, this order must divide the order of the group G by Lagrange s Theorem. The homomorphisms above also give an important cautionary example! 1

2 Example 1.2 (Image subgroups are not necessarily normal). Let G = Σ 3 and a = (12) G. Then φ a : Z G defined by φ a (n) = a n has image Im(φ a ) =< a > which is not normal in G. Before we study cyclic groups, we make a useful reformulation of the concept: Lemma 1.3. A group G is cyclic if and only if there is an epimorphism φ : Z G. Proof. = : If G =< x > is a cyclic group then the homomorphism defined in Example 1.1 is an epimorphism Z G. =: On the other hand if φ : Z G is an epimorphism then x = φ(1) is easily seen to generate G. Definition 1.4. We say H is a quotient group of G if there is an epimorphism G H. Now we are ready to prove the core facts about cyclic groups: Proposition 1.5. The following are facts about cyclic groups: (1) A quotient group of a cyclic group is cyclic. (2) Subgroups of cyclic groups are cyclic. (3) If G is a cyclic group then G is isomorphic to Z/dZ for a unique integer d 0. (Note that when d = 0, Z/0Z = Z). Thus a cyclic group is determined up to isomorphism by its order. Proof. (3): By Lemma 1.3, there is an epimorphism φ : Z G. Then the First Isomorphism Theorem shows that G = Z/dZ where dz = ker(φ), d 0. (1): Let G λ H and suppose G cyclic generated by x. It is easy to see that since λ is an epimorphism, λ(x) generates H and so H is cyclic also. (2): Now suppose G is cyclic and H G. Then by Lemma 1.3, we have an epimorphism Z φ G. Let S = φ 1 (H) then S Z and so is cyclic generated by some integer d 0. Then φ restricts to an epimorphism S H and so H is cyclic as it is a quotient of a cyclic group. 2

3 2 Invariant Subgroups We begin this section with yet another important example of a homomorphism: Theorem 2.1. Let G be a group and g G. We define γ g : G G by γ g (h) = ghg 1 for all h G. γ g is refered to as the conjugation map given by conjugation by g. Then γ g Aut(G) for all g G. We refer to the set {γ g g G} as the set of inner automorphisms of G and denote it Inn(G). Proof. The only thing we need to prove it that γ g is a bijective homomorphism of G to itself. We calculate γ g (hw) = ghwg 1 = ghg 1 gwg 1 = γ g (h)γ g (w) and hence γ g is an endomorphism of G. On the other hand γ g 1(γ g (h)) = γ g 1(ghg 1 ) = g 1 ghg 1 (g 1 ) 1 = h for all h G. Thus γ g 1 γ g = 1 G for all g G, and also γ g γ g 1 = 1 G by interchanging the roles of g and g 1. Thus each γ g is bijective and hence gives an automorphism of G. We next show that the correspondence g γ g itself is a homomorphism! Theorem 2.2 (Conjugation Action Homomorphism). Let G be a group, then the function Θ : G Aut(G) given by Θ(g) = γ g is a homomorphism with image Inn(G) and kernel Z(G) = {g G gh = hg for all h G}. Z(G) is called the center of G, it consists of the elements of G which commute with everything in G. Thus Inn(G) Aut(G), Z(G) G and G/Z(G) = Inn(G). Proof. We calculate: γ gg (h) = (gg )h(gg ) 1 = g(g hg 1 )g 1 = gγ g (h)g 1 = γ g (γ g (h)) for all g, g, h G. Thus we conclude γ gg = γ g γ g for all g, g G and hence Θ is a homomorphism. 3

4 It is clear that Im(Θ) = Inn(G) by definition. On the other hand g ker(θ) γ g = 1 G ghg 1 = h for all h G gh = hg for all h G g Z(G). The final line of the theorem then follows from the First Isomorphism Theorem and the general facts on images and kernels of homomorphisms. Definition 2.3. In the homework, you will in fact show that Inn(G) Aut(G). Thus we may define the quotient group Out(G) = Aut(G)/Inn(G), of outer automorphisms of G. Thus Out(G) measures in some sense the automorphisms of G which are not given by conjugation. Notice that if G is Abelian, all non-identity automorphisms of G have to be outer automorphisms. We look at some examples next, but first a basic lemma: Lemma 2.4 (Orders under homomorphisms). If f : G H is a homomorphism and x G has finite order then f(x) also has finite order and o(f(x)) o(x). If f : G H is an isomorphism then o(x) = o(f(x)) for all x G. Proof. Let x G have finite order o(x) = m 1. Then x m = e so applying f to both sides of this equation we fine f(x) m = f(x m ) = f(e) = e. Thus f(x) has finite order and o(f(x)) m as we desired to show. If f : G H is an isomorphism, it restricts to an isomorphism of < x > and < f(x) > and so o(x) = o(f(x)) in this case. We can now work out some examples: Example 2.5 (Automorphisms of Σ 3 ). As Σ 3 = 3! = 6, there are 6! = 720 bijections of Σ 3 to itself. We would like to decide which of these are automorphisms! We can compute the orders of elements of Σ 3 : Order1 : e Order2 : (12), (13), (23) Order3 : (123), (132) We have seen that {(12), (13)} generate Σ 3 and that automorphisms τ are uniquely determined by what they do on a generating set. Thus it suffices to 4

5 consider the possibilities for τ((12)) and τ((13)). Since (12) has order 2, we see that τ((12)) would have to have order 2 by Lemma 2.4. Thus there are 3 possibilities for τ((12)). Once we have chosen one of these, as τ((13)) must also have order 2, and τ is injective we have two possibilities left for τ((13)). Thus there are at most 3 2 = 6 automorphisms of Σ 3. On the other hand it is simple to check that Z(Σ 3 ) = {e} and so Inn(Σ 3 ) = Σ 3 /Z(Σ 3 ) has order 6. Thus we conclude that Aut(Σ 3 ) = Inn(Σ 3 ) = Σ 3 and that Out(Σ 3 ) = 1. Thus all automorphisms of Σ 3 are inner automorphisms. Notice also that only 6 out of the 720 bijections of Σ 3 to itself preserve the algebraic structure! We next consider subgroups invariant under various classes of endomorphisms of G: Definition 2.6. Let H G. We say that: (1) H is normal in G if γ g (H) H for all inner automorphisms γ g of G. We write H G in this case. (2) H is characteristic in G if τ(h) H for all automorphisms τ of G. We write H char G in this case. (3) H is fully invariant in G if λ(h) H for all endomorphisms λ of G. We write H f.i. G in this case. It is immediate that H f.i. G = H char G = H G. This is because Inn(G) Aut(G) End(G) and because if a subgroup is invariant under a bigger set of endomorphisms, it is also invariant for a subset. One thing we should check is that the definition of normal subgroup given in Definition 2.6 is the same as the one we previously gave! Note γ g (H) H for all g G is equivalent to ghg 1 H for all g G. This is in turn equivalent to ghg 1 = H for all g G. (This can be seen by replacing g with g 1.) Finally this is equivalent to gh = Hg for all g G which was our original definition of normality. Thus the two definitions coincide. The following is a basic proposition: Proposition 2.7. Let G be a group then: (1) K char H, H char G = K char G. 5

6 (2) K f.i. H, H f.i. G = K f.i. G. (3) K H, H G does not imply K G. Thus being a normal subgroup is not a transitive relation! Proof. We only prove (1). The proof of (2) is similar and is left to the reader. So suppose K char H, H char G and let τ Aut(G). Then τ restricts to an automorphism τ H of H as H is characteristic in G. (τ 1 will restrict to give the inverse automorphism.) Since K is characteristic in H, τ H (K) = τ(k) K. Thus we see that τ(k) K for all τ Aut(G) and so K char G. A counterexample for (3) will be done in the Homework. The proof for (1) and (2) does not carry over to (3) as the restriction of an inner automorphism of G to a subgroup H can yield an automorphism of H which is not inner. For example, in Σ 3, let H =< (123) > and let g = (12) Σ 3 H. Then conjugation by g, gives an inner automorphism γ g of Σ 3 which restricts to a nontrivial automorphism of H as H Σ 3. Moreover since H is Abelian, we see that γ g H is an outer automorphism of H. So restrictions of inner automorphisms need not be inner! We will give an example of a fully invariant subgroup but first a definition: Definition 2.8 (Commutators). Given a group G and x, y G we define the commutator [x, y] by [x, y] = xyx 1 y 1. It is simple to check that xy = [x, y]yx and so the commutator [x, y] measures the failure of x and y to commute. Note: [x, y] 1 = (xyx 1 y 1 ) 1 = yxy 1 x 1 = [y, x]. So the inverse of a commutator [x, y] is the commutator [y, x]. Example 2.9 (Commutator subgroup). Given a group G, we define the commutator subgroup G (sometimes denoted also by [G, G]) by G =< [x, y] x, y G >. 6

7 Thus G is generated by all the commutators in G. The typical element in G is a finite product of commutators [x 1, y 1 ][x 1, y 2 ]...[x k, y k ]. (we don t have to use inverses as the inverse of a commutator is itself a commutator). Note it is easy to see that G = {e} if and only if G is Abelian. Proposition 2.10 (Commutator Subgroups are Fully Invariant). Let G be a group, then G f.i. G and so in particular G G. The quotient group G/G is Abelian. Proof. Let f : G G be an endomorphism of G. Then we compute: f([x, y]) = f(xyx 1 y 1 ) = f(x)f(y)f(x) 1 f(y) 1 = [f(x), f(y)]. Thus f takes commutators to commutators and hence takes a finite product of commutators to a finite product of commutators. Hence f(g ) G. Since this holds for any endomorphism f, we see that G f.i G. Now we show G/G is Abelian. Let x, y G then it is simple to compute that [xg, yg ] = [x, y]g = G as [x, y] G. Thus xg, yg commute in G/G. Since xg, yg were arbitrary in G/G we conclude that G/G is Abelian. Theorem 2.11 (Abelianizations). Given a group G we define G ab, the Abelianization of G via G ab = G/G. We have seen previously that G ab is Abelian. In fact, it is the largest Abelian quotient of G. In other words, given an epimorphism G ψ A where A is Abelian, we may always find a homomorphism µ making the following diagram commute: G ψ A µ G/G = G ab In particular, A will be a quotient of G ab. Proof. This will follow from our fundamental factorization lemma once we can show G is contained in ker(ψ). Now ψ([x, y]) = [ψ(x), ψ(y)] = e as this final commutator lives in an Abelian group A. Thus all commutators of G lie in ker(ψ) and so it follows that G ker(ψ). The existance of a homomorphism µ making the diagram commute then follows from our fundamental factorization theorem. 7

### 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

### 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

### 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

### The Outer Automorphism of S 6

Meena Jagadeesan 1 Karthik Karnik 2 Mentor: Akhil Mathew 1 Phillips Exeter Academy 2 Massachusetts Academy of Math and Science PRIMES Conference, May 2016 What is a Group? A group G is a set of 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

### 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

### 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 =

### (Think: three copies of C) i j = k = j i, j k = i = k j, k i = j = i k.

Warm-up: The quaternion group, denoted Q 8, is the set {1, 1, i, i, j, j, k, k} with product given by 1 a = a 1 = a a Q 8, ( 1) ( 1) = 1, i 2 = j 2 = k 2 = 1, ( 1) a = a ( 1) = a a Q 8, (Think: three copies

### 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

### REU 2007 Discrete Math Lecture 2

REU 2007 Discrete Math Lecture 2 Instructor: László Babai Scribe: Shawn Drenning June 19, 2007. Proofread by instructor. Last updated June 20, 1 a.m. Exercise 2.0.1. Let G be an abelian group and A G be

### ABSTRACT ALGEBRA 1, LECTURE NOTES 5: HOMOMORPHISMS, ISOMORPHISMS, SUBGROUPS, QUOTIENT ( FACTOR ) GROUPS. ANDREW SALCH

ABSTRACT ALGEBRA 1, LECTURE NOTES 5: HOMOMORPHISMS, ISOMORPHISMS, SUBGROUPS, QUOTIENT ( FACTOR ) GROUPS. ANDREW SALCH 1. Homomorphisms and isomorphisms between groups. Definition 1.1. Let G, H be groups.

### 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.

### 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

### Extra exercises for algebra

Extra exercises for algebra These are extra exercises for the course algebra. They are meant for those students who tend to have already solved all the exercises at the beginning of the exercise session

### 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

### 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,...,

### 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

### 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

### 1 Chapter 6 - Exercise 1.8.cf

1 CHAPTER 6 - EXERCISE 1.8.CF 1 1 Chapter 6 - Exercise 1.8.cf Determine 1 The Class Equation of the dihedral group D 5. Note first that D 5 = 10 = 5 2. Hence every conjugacy class will have order 1, 2

### 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

### 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

### 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

### 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

### 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

### 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

### 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)

### 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

### 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

### 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

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

### 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)

### 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 SEP Solutions

Algebra SEP Solutions 17 July 2017 1. (January 2017 problem 1) For example: (a) G = Z/4Z, N = Z/2Z. More generally, G = Z/p n Z, N = Z/pZ, p any prime number, n 2. Also G = Z, N = nz for any n 2, since

### Notas de Aula Grupos Profinitos. Martino Garonzi. Universidade de Brasília. Primeiro semestre 2018

Notas de Aula Grupos Profinitos Martino Garonzi Universidade de Brasília Primeiro semestre 2018 1 Le risposte uccidono le domande. 2 Contents 1 Topology 4 2 Profinite spaces 6 3 Topological groups 10 4

### Math 121 Homework 4: Notes on Selected Problems

Math 121 Homework 4: Notes on Selected Problems 11.2.9. If W is a subspace of the vector space V stable under the linear transformation (i.e., (W ) W ), show that induces linear transformations W on W

### Math 429/581 (Advanced) Group Theory. Summary of Definitions, Examples, and Theorems by Stefan Gille

Math 429/581 (Advanced) Group Theory Summary of Definitions, Examples, and Theorems by Stefan Gille 1 2 0. Group Operations 0.1. Definition. Let G be a group and X a set. A (left) operation of G on X is

### 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

### A Note on Groups with Just-Infinite Automorphism Groups

Note di Matematica ISSN 1123-2536, e-issn 1590-0932 Note Mat. 32 (2012) no. 2, 135 140. doi:10.1285/i15900932v32n2p135 A Note on Groups with Just-Infinite Automorphism Groups Francesco de Giovanni Dipartimento

### 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

### Group Isomorphisms - Some Intuition

Group Isomorphisms - Some Intuition The idea of an isomorphism is central to algebra. It s our version of equality - two objects are considered isomorphic if they are essentially the same. Before studying

### 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

### Lecture 6 Special Subgroups

Lecture 6 Special Subgroups Review: Recall, for any homomorphism ϕ : G H, the kernel of ϕ is and the image of ϕ is ker(ϕ) = {g G ϕ(g) = e H } G img(ϕ) = {h H ϕ(g) = h for some g G} H. (They are subgroups

### 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

### COURSE SUMMARY FOR MATH 504, FALL QUARTER : MODERN ALGEBRA

COURSE SUMMARY FOR MATH 504, FALL QUARTER 2017-8: MODERN ALGEBRA JAROD ALPER Week 1, Sept 27, 29: Introduction to Groups Lecture 1: Introduction to groups. Defined a group and discussed basic properties

### 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

### 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

### 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

### 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

### book 2005/1/23 20:41 page 132 #146

book 2005/1/23 20:41 page 132 #146 132 2. BASIC THEORY OF GROUPS Definition 2.6.16. Let a and b be elements of a group G. We say that b is conjugate to a if there is a g G such that b = gag 1. You are

### 2. Intersection Multiplicities

2. Intersection Multiplicities 11 2. Intersection Multiplicities Let us start our study of curves by introducing the concept of intersection multiplicity, which will be central throughout these notes.

### 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.

### (1) Let G be a finite group and let P be a normal p-subgroup of G. Show that P is contained in every Sylow p-subgroup of G.

(1) Let G be a finite group and let P be a normal p-subgroup of G. Show that P is contained in every Sylow p-subgroup of G. (2) Determine all groups of order 21 up to isomorphism. (3) Let P be s Sylow

### On Conditions for an Endomorphism to be an Automorphism

Algebra Colloquium 12 : 4 (2005 709 714 Algebra Colloquium c 2005 AMSS CAS & SUZHOU UNIV On Conditions for an Endomorphism to be an Automorphism Alireza Abdollahi Department of Mathematics, University

### 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

### 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

### MATH 1530 ABSTRACT ALGEBRA Selected solutions to problems. a + b = a + b,

MATH 1530 ABSTRACT ALGEBRA Selected solutions to problems Problem Set 2 2. Define a relation on R given by a b if a b Z. (a) Prove that is an equivalence relation. (b) Let R/Z denote the set of equivalence

### Notes on Group Theory. by Avinash Sathaye, Professor of Mathematics November 5, 2013

Notes on Group Theory by Avinash Sathaye, Professor of Mathematics November 5, 2013 Contents 1 Preparation. 2 2 Group axioms and definitions. 2 Shortcuts................................. 2 2.1 Cyclic groups............................

### 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

### 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

### 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,

### CONJUGATION IN A GROUP

CONJUGATION IN A GROUP KEITH CONRAD 1. Introduction A reflection across one line in the plane is, geometrically, just like a reflection across any other line. That is, while reflections across two different

### 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

### 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

### PROBLEMS FROM GROUP THEORY

PROBLEMS FROM GROUP THEORY Page 1 of 12 In the problems below, G, H, K, and N generally denote groups. We use p to stand for a positive prime integer. Aut( G ) denotes the group of automorphisms of G.

### 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:

### Normal Automorphisms of Free Burnside Groups of Period 3

Armenian Journal of Mathematics Volume 9, Number, 017, 60 67 Normal Automorphisms of Free Burnside Groups of Period 3 V. S. Atabekyan, H. T. Aslanyan and A. E. Grigoryan Abstract. If any normal subgroup

### 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

### 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

### Stab(t) = {h G h t = t} = {h G h (g s) = g s} = {h G (g 1 hg) s = s} = g{k G k s = s} g 1 = g Stab(s)g 1.

1. Group Theory II In this section we consider groups operating on sets. This is not particularly new. For example, the permutation group S n acts on the subset N n = {1, 2,...,n} of N. Also the group

### 7 Semidirect product. Notes 7 Autumn Definition and properties

MTHM024/MTH74U Group Theory Notes 7 Autumn 20 7 Semidirect product 7. Definition and properties Let A be a normal subgroup of the group G. A complement for A in G is a subgroup H of G satisfying HA = G;

### 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

### 6. The Homomorphism Theorems In this section, we investigate maps between groups which preserve the groupoperations.

6. The Homomorphism Theorems In this section, we investigate maps between groups which preserve the groupoperations. Definition. Let G and H be groups and let ϕ : G H be a mapping from G to H. Then ϕ is

### SEMI-INVARIANTS AND WEIGHTS OF GROUP ALGEBRAS OF FINITE GROUPS. D. S. Passman P. Wauters University of Wisconsin-Madison Limburgs Universitair Centrum

SEMI-INVARIANTS AND WEIGHTS OF GROUP ALGEBRAS OF FINITE GROUPS D. S. Passman P. Wauters University of Wisconsin-Madison Limburgs Universitair Centrum Abstract. We study the semi-invariants and weights

### Pseudo Sylow numbers

Pseudo Sylow numbers Benjamin Sambale May 16, 2018 Abstract One part of Sylow s famous theorem in group theory states that the number of Sylow p- subgroups of a finite group is always congruent to 1 modulo

### 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

### 3. G. Groups, as men, will be known by their actions. - Guillermo Moreno

3.1. The denition. 3. G Groups, as men, will be known by their actions. - Guillermo Moreno D 3.1. An action of a group G on a set X is a function from : G X! X such that the following hold for all g, h

### 6 More on simple groups Lecture 20: Group actions and simplicity Lecture 21: Simplicity of some group actions...

510A Lecture Notes 2 Contents I Group theory 5 1 Groups 7 1.1 Lecture 1: Basic notions............................................... 8 1.2 Lecture 2: Symmetries and group actions......................................

### 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

### 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

### Definition : Let G be a group on X a set. A group action (or just action) of G on X is a mapping: G X X

1 Frank Moore Algebra 901 Notes Professor: Tom Marley Group Actions on Sets Definition : Let G be a group on X a set. A group action (or just action) of G on X is a mapping: G X X Such the following properties

### Some algebraic properties of. compact topological groups

Some algebraic properties of compact topological groups 1 Compact topological groups: examples connected: S 1, circle group. SO(3, R), rotation group not connected: Every finite group, with the discrete

### MATH 323, Algebra 1. Archive of documentation for. Bilkent University, Fall 2014, Laurence Barker. version: 17 January 2015

Archive of documentation for MATH 323, Algebra 1 Bilkent University, Fall 2014, Laurence Barker version: 17 January 2015 Source file: arch323fall14.tex page 2: Course specification. page 4: Homeworks and

### SPRING BREAK PRACTICE PROBLEMS - WORKED SOLUTIONS

Math 330 - Abstract Algebra I Spring 2009 SPRING BREAK PRACTICE PROBLEMS - WORKED SOLUTIONS (1) Suppose that G is a group, H G is a subgroup and K G is a normal subgroup. Prove that H K H. Solution: We

### 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

### 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

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

### EXERCISES ON THE OUTER AUTOMORPHISMS OF S 6

EXERCISES ON THE OUTER AUTOMORPHISMS OF S 6 AARON LANDESMAN 1. INTRODUCTION In this class, we investigate the outer automorphism of S 6. Let s recall some definitions, so that we can state what an outer

### 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

### Math 594. Solutions 5

Math 594. Solutions 5 Book problems 6.1: 7. Prove that subgroups and quotient groups of nilpotent groups are nilpotent (your proof should work for infinite groups). Give an example of a group G which possesses

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

### φ(a + b) = φ(a) + φ(b) φ(a b) = φ(a) φ(b),

16. Ring Homomorphisms and Ideals efinition 16.1. Let φ: R S be a function between two rings. We say that φ is a ring homomorphism if for every a and b R, and in addition φ(1) = 1. φ(a + b) = φ(a) + φ(b)

### Maximal non-commuting subsets of groups

Maximal non-commuting subsets of groups Umut Işık March 29, 2005 Abstract Given a finite group G, we consider the problem of finding the maximal size nc(g) of subsets of G that have the property that no

### 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

### 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

### 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.

### Written Homework # 1 Solution

Math 516 Fall 2006 Radford Written Homework # 1 Solution 10/11/06 Remark: Most of the proofs on this problem set are just a few steps from definitions and were intended to be a good warm up for the course.

### 2. Modules. Definition 2.1. Let R be a ring. Then a (unital) left R-module M is an additive abelian group together with an operation

2. Modules. The concept of a module includes: (1) any left ideal I of a ring R; (2) any abelian group (which will become a Z-module); (3) an n-dimensional C-vector space (which will become both a module

### On The Number Of Distinct Cyclic Subgroups Of A Given Finite Group

BearWorks Institutional Repository MSU Graduate Theses Spring 2016 On The Number Of Distinct Cyclic Subgroups Of A Given Finite Group Joseph Dillstrom As with any intellectual project, the content and