# 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.

Size: px
Start display at page:

Download "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."

Transcription

1 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 homomorphism. Proof: Elements of G H are all pairs {(g, h) g G, h H} operation is componentwise elements of H G are all pairs {(h, g) g G, h H} operation is componentwise f((g 1, h 1 ) + (g 2, h 2 )) = f(g 1 + g 2, h 1 + h 2 ) = (h 1 + h 2, g 1 + g 2 ) f(g 1, h 1 ) + f(g 2, h 2 ) = (h 1, g 1 ) + (h 2, g 2 ) = (h 1 + h 2, g 1 + g 2 ) Therefore f((g 1, h 1 ) + (g 2, h 2 )) = f(g 1, h 1 ) + f(g 2, h 2 ) hence f is a homomorphism. i. Find Ker(f). identity of G H is the pair (e G, e H )! identity of H G is the pair (e H, e G )! Let (a, b) Ker(f). Then f(a, b) is the identity in H G. So f(a, b) = (e H, e G ). But f(a, b) = (b, a) by definition of f. Therefore (e H, e G ) = (b, a). Therefore e H = b and e G = a. So (a, b) = (e G, e H ). Hence Ker(f) = {(e G, e H )} ii. Find Im(f). Im(f) = H G. You have to prove this! (b) Prove that the map i : G G H defined as i(g) = (g, e h ) is a group homomorphism (e H is the identity element in H). i(g 1 + g 2 ) = (g 1 + g 2, e H ) i(g 1 ) + i(g 2 ) = (g 1, e H ) + (g 2, e H ) = (g 1 + g 2, e H + e H ) = (g 1 + g 2, e H ) Therefore i(g 1 + g 2 ) = i(g 1 ) + i(g 2 ) i. Find Ker(i). Ker(i) = {e G }. You have to prove this! ii. Find Im(i). Im(i) = G {e H } = {(g, e H ) g G} < G H 1

2 (c) Prove that the map p : G H G defined as p(g, h) = g is a group homomorphism. i. Find Ker(p). ii. Find Im(p). 2. Describe all Abelian groups G, up to isomorphism, such that: (a) G = 16 Z 16, Z 8 Z 2, Z 4 Z 4, Z 4 Z 2 Z 2, Z 2 Z 2 Z 2 Z 2 (b) G = 16 and G has no elements of order Z 16 has order 16. (a, b) = lcm( a, b ) for (a, b) A B a Z 8 = a = 1, 2, 4, 8 a Z 4 = a = 1, 2, 4. a Z 2 = a = 1, 2. lcm(8, 2) = 8 lcm(4, 2) = lcm(4, 4) = lcm(2, 4) = 4 lcm(2, 2) = lcm(2, 2, 2) = 2 lcm(k, 1) = k The only possible orders of the elements in the following groups are: 8,4,2,1. Z 8 Z 2, Z 4 Z 4, Z 4 Z 2 Z 2, Z 2 Z 2 Z 2 Z 2 (c) G = 16 and G has no elements of order 8. 2 Z 16 has order 8 (1, 0) Z 8 Z 2 has order 8. The only possible orders of the elements in the following groups are: 4,2,1. Z 4 Z 4, Z 4 Z 2 Z 2, Z 2 Z 2 Z 2 Z 2 (d) G = 16 and all non-identity elements of G have order 2.. Z 2 Z 2 Z 2 Z 2 2

3 (e) G = = G (2) = Z 8 or Z 4 Z 2 or Z 2 Z 2 Z 2 G (3) = Z 9 or Z 3 Z 3 G (5) = Z 5 G is isomorphic to one of the following 6 groups: Z 8 Z 9 Z 5 Z 4 Z 2 Z 9 Z 5 Z 2 Z 2 Z 2 Z 9 Z 5 Z 8 Z 3 Z 3 Z 5 Z 4 Z 2 Z 3 Z 3 Z 5 Z 2 Z 2 Z 2 Z 3 Z 3 Z 5 (f) G = 12 but G does not have elements of order 12. If G = 12 = and G is Abelian, then G is isomorphic to one of the following: Z 4 Z 3, Z 2 Z 2 Z 3 1 Z 4 has order 4. 1 Z 3 has order 3. (1, 1) Z 4 Z 3 has order (1, 1) = lcm(4, 3) = 12 G = Z 2 Z 2 Z 3 (g) G = 18 but G is not cyclic. G = 18 = So G is isomorphic to a product of an Abelian group G (2) of order 2 and an Abelian group G (3) of order 3 2. The only possibility for G (2) is: Z 2. The only possibilities for G (3) are: Z 9 and Z 3 Z 3. So G is isomorphic to one of: Z 2 Z 9 or Z 2 Z 3 Z 3. External product of cyclic groups of relatively prime orders is isomorphic to a cyclic group with order equal to the product of orders (done in class). So Z 2 Z 9 = Z18 which is cyclic. Therefore there is only one Abelian non-cyclic group of order 18 (up to isomorphism!): Z 2 Z 3 Z Let G be a group acting on a set X. Main formulas and properties to use: X = disjoint O x O x O x divides G O x = 1 if and only if x is a fixed point of the action of G on X. 3

4 (a) If G = 11 and X = 12 prove that the action has at least one fixed point. Proof: Since G = 11, the orbits can have 1 or 11 elements. Since X = 12 we have 12 = X = disjoint O x O x. So, we have either 12 = 11+1 or 12 = These are the only possibilities and in both cases there is at least one x X such that the orbit O x = 1 has only one element, hence G fixes that point, i.e. x is a fixed point of the action. (b) If G = 11 and X = 10 prove that G(x) = x for all x X. Proof: Since G = 11, the orbits can have 1 or 11 elements. Since X = 10 we have 10 = X = disjoint O x O x. So, we have 10 = So for each x X the orbit O x = 1 has only one element, hence G fixes that point, i.e. gx = x for all g G, or G(x) = x. Hence for each x X we have G(x) = x. (c) If G = 12 and X = 11 prove that the action has at least 3 orbits. (d) If G = 16 and X = 155 prove that the action has at least one fixed point. (e) If G = p 2 where p is a prime and X = q, q a prime q > p then G has at least one fixed point. 4. Let G X ϕ X be an action of group G on the set X = G given by conjugation, i.e. ϕ(g, x) = gxg 1. Suppose x is a fixed point of this action. Prove that x Z(G), i.e. x is in the center of G. Proof: Let x be a fixed point of the action of the group G on the set X = G, given by ϕ(g, x) = gxg 1. If x is fixed point then ϕ(g, x) = gxg 1 = x for all g G. So gxg 1 = x for all g G. Multiply on the right by g and get: So gx = xg for all g G. Therefore x Z(G), the center of G. (Def. Z(G) = {x G gx = xg for all g G}.) 5. Let G X ϕ X be an action of group G on the set X = {subgroups of G} given by conjugation, i.e. ϕ(g, H) = ghg 1. Suppose K is a fixed point of this action. Prove that K is a normal subgroup of G. 6. Let f : G G be a group homomorphism. Prove: If Ker(f) = e then f is a one-to-one map. Proof: Assume Ker(f) = {e}. WTS f is one-to-one. Suppose f(x 1 ) = f(x 2 ). WTS x 1 = x 2. 4

5 Use two facts about homomorphisms, which we did in class: f(ab) = f(a)f(b) this is just definition f(a 1 ) = (f(a)) 1 mentioned many times in class (for homomorphisms) f(x 1 x 1 2 ) = f(x 1 ) f(x 1 2 ) by the first property = f(x 1 ) f(x 1 2 ) = f(x 1 ) (f(x 2 )) 1 by the second property = f(x 1 ) (f(x 1 )) 1 by assumption that f(x 1 ) = f(x 2 ) = e G since f(x 1 ) and (f(x 1 )) 1 are inverses of each other. Therefore f(x 1 x 1 2 ) = e G Therefore x 1 x 1 2 Ker(f) = {e}. So x 1 x 1 2 = e. Now multiply the equation by x 2 on the right. Therefore (x 1 x 1 2 ) x 2 = e x 2. Apply associative law, inverse and identity property. So x 1 = x 2. Therefore f is one-to-one. 7. Let f : G G be a group homomorphism. Prove: Ker(f) is a normal subgroup of G. 8. If G = 18 and a G, what are all possible orders of a? Answer: 1,2,3,6,9, If G = 17 and a G, what are all possible orders of a? Answer: 1, If G = 23 and a G, a e, what are all possible orders of a? Answer: Let G be a group of order G = 35. (a) How many 5-Sylow subgroups does G have? r 5 = #{5-Sylow subgroups} G = 35 = r 5 = 1, 5, 7, 35 r 5 = #{5-Sylow subgroups} = 1(mod5) = r 5 = 1, 6, 11, 16, 21, 26, 31 r 5 = 1. Therefore there is exactly one 5-Sylow subgroup. (b) Let P 5 be a 5-Sylow subgroup of G. How many elements does P 5 have? P 5 = 5 (c) How many 7-Sylow subgroups does G have? r 7 = #{7-Sylow subgroups} G = 35 = r 7 = 1, 5, 7, 35 r 7 = #{7-Sylow subgroups} = 1(mod7) = r 7 = 1, 8, 15, 22, 29 r 7 = 1. Therefore there is exactly one 7-Sylow subgroup. (d) Let P 7 be a 7-Sylow subgroup of G. Find P 7. P 7 = 7 5

6 12. Find ALL Sylow subgroups of Z 12. Prove that these are all. r 2 = #{2-Sylow subgroups} G = 12 = r 2 = 1, 2, 3, 4, 6, 12 r 2 = #{2-Sylow subgroups} = 1(mod2) = r 2 = 1, 3, 5, 7, 9, 11 r 2 = 1 or 3. Notice, since Z 12 is Abelian, all subgroups are normal and since all 2-Sylow subgroups are conjugate to each other, it follows that gp 2 g 1 = P 2 for all g G. Hence there is only one 2-Sylow subgroup P 2. P 2 = 4 and P 2 = {3, 6, 9, 0}. Similarly, there is exactly one 3-Sylow subgroup P 3 = {4, 8, 0}. 13. Determine the Sylow subgroups of the alternating group A 4 (the even permutations of {1, 2, 3, 4}. A 4 = 12 = A 4 is not Abelian, so we really need to find numbers of p-sylow subgroups for p = 2 and p = 3 r 2 = #{2-Sylow subgroups} G = 12 = r 2 = 1, 2, 3, 4, 6, 12 r 2 = #{2-Sylow subgroups} = 1(mod 2) = r 2 = 1, 3, 5, 7, 9, 11 r 2 = 1 or 3. P 2 = 4 and P 2 = {(12)(34), (13)(24), (14)(23), (1)} r 3 = #{3-Sylow subgroups} G = 12 = r 3 = 1, 2, 3, 4, 6, 12 r 3 = #{3-Sylow subgroups} = 1(mod 3) = r 3 = 1, 4, 7, 10 r 2 = 1 or 4. {P (1) 3 = (123), P (2) 3 = (124), P (3) 3 = (134) P (4) 3 = (234) } 6

### 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

### 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

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

### 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

### 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

### 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

### 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,...,

### MAT 150A, Fall 2015 Practice problems for the final exam

MAT 150A, Fall 2015 Practice problems for the final exam 1. Let f : S n! G be any homomorphism (to some group G) suchthat f(1 2) = e. Provethatf(x) =e for all x. Solution: The kernel of f is a normal subgroup

### 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

### 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

### LECTURES 11-13: CAUCHY S THEOREM AND THE SYLOW THEOREMS

LECTURES 11-13: CAUCHY S THEOREM AND THE SYLOW THEOREMS Recall Lagrange s theorem says that for any finite group G, if H G, then H divides G. In these lectures we will be interested in establishing certain

### 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

### 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

### 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.

### 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

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

### 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

### Math 120: Homework 6 Solutions

Math 120: Homewor 6 Solutions November 18, 2018 Problem 4.4 # 2. Prove that if G is an abelian group of order pq, where p and q are distinct primes then G is cyclic. Solution. By Cauchy s theorem, G has

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

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

### 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

### 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 =

### Modern Algebra I. Circle the correct answer; no explanation is required. Each problem in this section counts 5 points.

1 2 3 style total Math 415 Please print your name: Answer Key 1 True/false Circle the correct answer; no explanation is required. Each problem in this section counts 5 points. 1. Every group of order 6

### 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

### x 2 = xn xn = x 2 N = N = 0

Potpourri. Spring 2010 Problem 2 Let G be a finite group with commutator subgroup G. Let N be the subgroup of G generated by the set {x 2 : x G}. Then N is a normal subgroup of G and N contains G. Proof.

### 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

### Homework #5 Solutions

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

### 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

### (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

### 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.

### 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

### 15 Permutation representations and G-sets

15 Permutation representations and G-sets Recall. If C is a category and c C then Aut(c) =the group of automorphisms of c 15.1 Definition. A representation of a group G in a category C is a homomorphism

### The Class Equation X = Gx. x X/G

The Class Equation 9-9-2012 If X is a G-set, X is partitioned by the G-orbits. So if X is finite, X = x X/G ( x X/G means you should take one representative x from each orbit, and sum over the set of representatives.

### 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

### 1 Finite abelian groups

Last revised: May 16, 2014 A.Miller M542 www.math.wisc.edu/ miller/ Each Problem is due one week from the date it is assigned. Do not hand them in early. Please put them on the desk in front of the room

### ALGEBRA HOMEWORK SET 2. Due by class time on Wednesday 14 September. Homework must be typeset and submitted by as a PDF file.

ALGEBRA HOMEWORK SET 2 JAMES CUMMINGS (JCUMMING@ANDREW.CMU.EDU) Due by class time on Wednesday 14 September. Homework must be typeset and submitted by email as a PDF file. (1) Let H and N be groups and

### 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

### Properties of Homomorphisms

Properties of Homomorphisms Recall: A function φ : G Ḡ is a homomorphism if φ(ab) = φ(a)φ(b) a, b G. Let φ : G Ḡ be a homomorphism, let g G, and let H G. Properties of elements Properties of subgroups

### 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

### 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

### MAT1100HF ALGEBRA: ASSIGNMENT II. Contents 1. Problem Problem Problem Problem Problem Problem

MAT1100HF ALEBRA: ASSINMENT II J.A. MRACEK 998055704 DEPARTMENT OF MATHEMATICS UNIVERSITY OF TORONTO Contents 1. Problem 1 1 2. Problem 2 2 3. Problem 3 2 4. Problem 4 3 5. Problem 5 3 6. Problem 6 3 7.

### Name: Solutions Final Exam

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

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

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

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

### 13 More on free abelian groups

13 More on free abelian groups Recall. G is a free abelian group if G = i I Z for some set I. 13.1 Definition. Let G be an abelian group. A set B G is a basis of G if B generates G if for some x 1,...x

### 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

### 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

### Direction: You are required to complete this test within 50 minutes. Please make sure that you have all the 10 pages. GOOD LUCK!

Test 3 November 11, 2005 Name Math 521 Student Number Direction: You are required to complete this test within 50 minutes. (If needed, an extra 40 minutes will be allowed.) In order to receive full credit,

### 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

### Note that a unit is unique: 1 = 11 = 1. Examples: Nonnegative integers under addition; all integers under multiplication.

Algebra fact sheet An algebraic structure (such as group, ring, field, etc.) is a set with some operations and distinguished elements (such as 0, 1) satisfying some axioms. This is a fact sheet with definitions

### May 6, Be sure to write your name on your bluebook. Use a separate page (or pages) for each problem. Show all of your work.

Math 236H May 6, 2008 Be sure to write your name on your bluebook. Use a separate page (or pages) for each problem. Show all of your work. 1. (15 points) Prove that the symmetric group S 4 is generated

### 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

### Name: Solutions Final Exam

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

### 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

### 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.

### 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

### 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

### MAT301H1F Groups and Symmetry: Problem Set 2 Solutions October 20, 2017

MAT301H1F Groups and Symmetry: Problem Set 2 Solutions October 20, 2017 Questions From the Textbook: for odd-numbered questions, see the back of the book. Chapter 5: #8 Solution: (a) (135) = (15)(13) is

### 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.

### 1. Group Theory Permutations.

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

### Abstract Algebra, HW6 Solutions. Chapter 5

Abstract Algebra, HW6 Solutions Chapter 5 6 We note that lcm(3,5)15 So, we need to come up with two disjoint cycles of lengths 3 and 5 The obvious choices are (13) and (45678) So if we consider the element

### 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.

### xy xyy 1 = ey 1 = y 1 i.e.

Homework 2 solutions. Problem 4.4. Let g be an element of the group G. Keep g fixed and let x vary through G. Prove that the products gx are all distinct and fill out G. Do the same for the products xg.

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

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

### MAT301H1F Groups and Symmetry: Problem Set 3 Solutions November 18, 2017

MAT301H1F Groups and Symmetry: Problem Set 3 Solutions November 18, 2017 Questions From the Textbook: for odd-numbered questions see the back of the book. Chapter 8: #8 Is Z 3 Z 9 Z 27? Solution: No. Z

### 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

### 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

### Johns Hopkins University, Department of Mathematics Abstract Algebra - Spring 2009 Midterm

Johns Hopkins University, Department of Mathematics 110.401 Abstract Algebra - Spring 2009 Midterm Instructions: This exam has 8 pages. No calculators, books or notes allowed. You must answer the first

### 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

### 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

### 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)

### 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

### 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

### Converse to Lagrange s Theorem Groups

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

### 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

### 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 =

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

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

### 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

### Kevin James. p-groups, Nilpotent groups and Solvable groups

p-groups, Nilpotent groups and Solvable groups Definition A maximal subgroup of a group G is a proper subgroup M G such that there are no subgroups H with M < H < G. Definition A maximal subgroup of a

### QUALIFYING EXAM IN ALGEBRA August 2011

QUALIFYING EXAM IN ALGEBRA August 2011 1. There are 18 problems on the exam. Work and turn in 10 problems, in the following categories. I. Linear Algebra 1 problem II. Group Theory 3 problems III. Ring

### 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

### 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

### The Symmetric Groups

Chapter 7 The Symmetric Groups 7. Introduction In the investigation of finite groups the symmetric groups play an important role. Often we are able to achieve a better understanding of a group if we can

### (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

### HOMEWORK Graduate Abstract Algebra I May 2, 2004

Math 5331 Sec 121 Spring 2004, UT Arlington HOMEWORK Graduate Abstract Algebra I May 2, 2004 The required text is Algebra, by Thomas W. Hungerford, Graduate Texts in Mathematics, Vol 73, Springer. (it

### 38 Irreducibility criteria in rings of polynomials

38 Irreducibility criteria in rings of polynomials 38.1 Theorem. Let p(x), q(x) R[x] be polynomials such that p(x) = a 0 + a 1 x +... + a n x n, q(x) = b 0 + b 1 x +... + b m x m and a n, b m 0. If b m

### 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

### 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,*)

### Maximal finite subgroups of Sp(2n, Q)

Maximal finite subgroups of Sp(2n, Q) Markus Kirschmer RWTH Aachen University Groups, Rings and Group-Rings July 12, 2011 Goal Characterization Goal: Classify all conjugacy classes of maximal finite symplectic

### 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

### 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.

### Fall 2014 Math 122 Midterm 1

1. Some things you ve (maybe) done before. 5 points each. (a) If g and h are elements of a group G, show that (gh) 1 = h 1 g 1. (gh)(h 1 g 1 )=g(hh 1 )g 1 = g1g 1 = gg 1 =1. Likewise, (h 1 g 1 )(gh) =h

### GROUP ACTIONS EMMANUEL KOWALSKI

GROUP ACTIONS EMMANUEL KOWALSKI Definition 1. Let G be a group and T a set. An action of G on T is a map a: G T T, that we denote a(g, t) = g t, such that (1) For all t T, we have e G t = t. (2) For all

### 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.

### 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,

### 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

### 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 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