# B Sc MATHEMATICS ABSTRACT ALGEBRA

Size: px
Start display at page:

Transcription

1 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 (a) (b) a b a b a b c, where c is the smallest integer greater than both a and b. (c) (d) a b c, where c is at least more than a b. a b c, where c is the largest integer less than the product of a and b. () Let Z be a set of integers, then under ordinary multiplication (Z,*) is (a) Group (b) Semi group (c) Monoid () If b and c are the inverses of some element a in a group G then : (a) b c (b) b c (c) b kc for some k N Abstract Algebra

2 (). How many non-trivial proper subgroups does the Klein -group have (a) 0 (b) (c) (d) () Under which of the following binary operation, the set of positive integers is closed (a) a b a ( b) a b a b b (c) a b a b ( d) a b ab () On Q, which of the following does not define a binary operation ( a) a b a b (b) a b a b (c) a b ab () Let be defined on Q as a b ab. Then identity for this binary operation is: (a) ( b) (c) 0 () The order of the subgroup of Z generated by the element is : ( a) (b) (c) (d) (9) Which of the following is a group (a) All n n diagonal matrices under addition (b) All n n diagonal matrices under matrix multiplication (c) All n n upper triangular matrices under matrix multiplication (0) Let G be a group and let a b c e for a, b, c G. Then b c a equals : (a) a ( b) e (c) b () Let be the binary operation defined on Q (a) a Then inverse of the element a is: (b) (c) a Abstract Algebra as a b ab. a () Value of the product ( ) ( - ) in Z is : (a) - (b) (c) 0 () Which of the following set addition is not a binary operation (a) Complex numbers (b) Real numbers (c) Non zero real numbers (d) Integers () How many binary operations can be defined on a set S with exactly n elements (a) (b) (c) (d)

3 () How many commutative binary operations can be defined on a set S with exactly n elements n( n) (a) n (b) ) (c) (d) () Which of the following are true () ( ) is a group. () is a group. () (Z, +) is a group. () (Z,. ) is a group. (a) & (b) Only (c) & (d) All () Which of the following are true () A group may have more than one identity element () Any two groups of three elements are isomorphic () Every group of at most three elements is abelian (a) & (b) & (c) & (d) All n n( n) () Let G= {, -, i, -i} where, be a set of four elements. Which of the following is a binary operation on G () a b a b () a b a.b (a) Only (b) Only (c) Both (9) Suppose a group G has an element x such that ax x for all a, then the number of elements in G is (a) (b) (c) (d) (0) Let G= C* ( non zero complex numbers). Which of the following are subgroups of G () All purely imaginary complex numbers under multiplication. () All complex numbers with absolute value under multiplication. (a) Only (b) Both (c) Only () Which of the following are true () ( ) is a subgroup of ( ). () is a subgroup of ( ) () ( ) is a subgroup of (R, +) (a) Only (b) Only (c) Only (d) All n n n () In a group G, a b a b for all a, b G. This statement is: (a) Always true (c) True if G is a multiplicative group (b) True if G is finite (d) True if G is abelian Abstract Algebra

4 () Which of the following is not true ( a) Z, is a proper subgroup of R, ( b) Q, is a proper subgroup of R,. (c) (d) Z, is a proper subgroup of Z, Q, is a proper subgroup of R, () Let k be a fixed positive integer and let kz kn n Z kz is a group which is isomorphic to R, /. Then : (a), (b) kz is not a group with respect to addition (c) kz, is a group which is isomorphic to Z, () The Klein - group is isomorphic to (a) Z Z (b) Z Z (c) Z () If G is a group of order four. Which of the following are not true () There exist a such that () There exist a such that () G is abelian (a) Only (b) & (c) Only (d) All () Consider with the binary operation m n = m + n. Which of the following are true () is commutative () is associative (a) Only (b) Only (c)all () Let (G, ) is a group. Which of the following are not always true () ( () = for all a, b in G () for all a, b in G (a) Only (b) Only (c) Only (d) & (9) Which of the following is a non-abelian group with the property that each proper subgroup is abelian (a) S (b) Z (c) S (0) The center of an abelian group G is: a) {e} b) G c) A cyclic subgroup d) None of these Abstract Algebra

5 () Let G be a cyclic group of order. Then the number of elements g G such that G = < g > is : ( a) (b) (c) (d) () Which of the following is true (a) Every cyclic group has a unique generator (b) In a cyclic group, every element is a generator (c) Every cyclic group has at least two generators () If a, b are elements of a group G of order m then order of ab and ba are: (a) same (b) Equal to m (c) unequal () Let G be a cyclic group of order. Then the number of elements g G such that G = < g > is : (a) (b) (c) (d) () How many elements are there on the cyclic subgroup of Z0 generated by (a) ( b) ( c) () Number of elements in the cyclic subgroup of the group of non-zero complex i numbers under multiplication, generated by, is: ( a) (b) (c) () Which of the following is a cyclic group with only one generator (a) Z (b) ( Z, ) ( c) Klein- group () How many generators are there for an infinite cyclic group (a) (b) (c) (d) Infinitely many (9) Which of the following is an example for a cyclic group with four generators (a) Z (b) Z (c) Z 9 (d) Z, (0) The least common multiple of r, s Z is rs if and only if: (a) r, s are relatively prime (b) One of them is a prime (c) Both are prime Abstract Algebra

6 () Which of the following is not true (a) Every cyclic group of order > has at least two distinct generators (b) If every proper subgroup of G is cyclic, then G itself is cyclic (c) Every sub group of a cyclic group is cyclic () Which of the following is a true statement (a) Every abelian group is cyclic (b) All generators of Z 0 are prime numbers (c) If G,G are groups, then G G is a group () How many elements are there in the subgroup generated by,, of the group Z (a) (b) (c) ( d) None of these () If, the order of the cyclic subgroup generated by is: (a) (b) (c) () Let. Then 00 (a) equals: (b) (c) () Let, then the orbit of under this permutation is: (a),,, (b),,,, (c),,, () Which of the following is a permutation function on R x (a) f ( x) x (b) f ( x) e (c) f ( x) x x x (d) f ( x) x (). Which of the following is true (a) Every function is a permutation if and only if it is one to one. (b) The symmetric group S is cyclic (c) The symmetric group S n is not cyclic for any n. (d) Every function from a finite set onto itself must be one to one. Abstract Algebra

7 Abstract Algebra (9) How many orbits are there for the permutation Z Z :, defined by ) ( n n (a) (b) (c) (d) infinitely many (0). How many orbits are there for the permutation Z Z :, defined by ) ( n n (a) (b) (c) (d) infinitely many (). How many elements are there in the orbit of under the permutation (a) (b) (c) (d) () The product,,,,, in S equals: (a) (b) (c) (d) () Which of the following represents (a),,,, (b),,,,, (c),,,, () Which of the following is an even permutation (a) (b) (c) (d) () Which of the following is not true (a) Every cycle is a permutation (b) A is a commutative group (c) A has 0 elements. () The order of (, 0 ) in Z Z is: (a) 9 ( b) (c)

8 () The order of (, 0, 9 ) in Z Z Z is: ( a) 0 (b) 0 (c) 0 () What is the largest possible order of a cyclic subgroup of Z Z (a) 0 (b) 0 (c) 0 (9) The cyclic subgroup of Z generated by has order (a) (b) ( c) 9 (0) The element (, ) of Z Z has order (a) (b) (c) (). Suppose G is a group of order 9. Then which of the following is true (a) G is not abelian (b) G has no subgroup other than (e) and G (c) There is a group H of order 9 which is not isomorphic to G (d) G is a subgroup of a group of order 0 () In a non-abelian group the element a has order 0. Then the order of a is: (a) (b) (c) (d) 9 () Let G be a group and g be fixed element of G. Then the map i g i g x gxg x G is : defined by (a) a one-one map on G, but not onto (b) an isomorphism of G with a proper subgroup of G (c) a bijective map from G to G, but not satisfies the homomorphism property (d) an isomorphism of G with itself. () If G is a group with no proper nontrivial subgroups, then G is: ( a) Not abelian (b) Cyclic (c) Isomorphic to Z, (d) An infinite group () How many automorphisms are there on the group Z (a) (b) (c) (d) None () How many subgroups are there for the group Z (a) (b) (c) () Which of the following group has no proper non trivial subgroups (a) Z ( b) Z (c) Z, (d) Klein- group Abstract Algebra

9 () If, then the smallest subgroup containing (a) (b) (c) (d) has order: (9) How many cosets are there of the subgroup Z of Z (a) (b) (c) (d) (0) Which of the following is not a coset of the subgroup of Z (a),,9 (b),,0 (c) 0,, ( d) None of these () The index of the subgroup in Z (a) (b) (c) (d) () If,,,,, then the index of in S (a) 0 (b) 0 ( c) 0 () If,,,,, then the index of in S (a) 0 (b) (c) () Which of the following is not true (a) Every subgroup of every group has left cosets (b) A subgroup of a group is a left coset of itself (c) A is of index in S for n > n n () Let G be a group having elements a and b such that O( a), O( b) and a b ba. Then O(ab) equals: ( a) (b) (c) (d) () The number of elements of order in the group Z, is: (a) (b) (c) (d) () Let G be any group and G G g g : ; ( ), g G (a) is a homomorphism (b) is a homomorphism if g is a finite group (c) is a homomorphism if g is abelian (d) is a homomorphism if and only if g is cyclic. Then: Abstract Algebra 9

10 (). Let : Z Z be a homomorphism such that ( ). Then Ker( ) equals: (a) Z (b) Z (c) Z (9) What is the value of (), where : Z Z be a homomorphism such that ( ) (a) 0 (b) (c) (d) (0) How many homomorphisms are there of Z onto Z (a) (b) (c) None (d) Infinitely many () Which of the following is true (a) Every cyclic group has prime order (b) Every abelian group is cyclic (c) Every group of prime order is cyclic () The sign of an even permutation is + and that of an odd permutation is -. Let sgn n : S n, be the homomorphism of Sn onto the multiplicative group defined by s gn n ( ) = sign of. Then kernel of is: (a) S n (b) An (c) { Identity permutation } () How many elements are there in the factor group Z / (a) ( b) ( c) (d) () Which of the following is true (a) Every homomorphism is a one to one map ( b) A homomorphism may have an empty kernel (c) For any two groups G and K, there exists a homomorphism of G into K (d) For any two groups G and K, there exists an isomorphism of G onto K () How many elements of finite order are there in Z Z Z (a) ( b) ( c) (d) Infinitely many () Let U be the multiplicative group of complex numbers of modulus and let xi : R, be the homomorphism defined as ( x) e, x R. Then U the kernel of is: (a) R (b) Z (c) Z (d) Abstract Algebra 0

11 () If G is an infinite cyclic group, then how many generators are there for the group G (a) Exactly two (b) At least two (c) Infinitely many (d) Only one () A cyclic group with only one generator can have at most elements (a) (b) (c) (d) infinitely many (9) The number of generators of the cyclic group of order is. (a) (b) (c) (d) (90) Which of the following is a homomorphism of groups (a) : R Z ; ( x) the greatest integer x. (b) : R R ; ( x) x under addition (c) Z Z ; ( ) the remainder of x when divided by (d) : 9 x : R R ; ( x ) x under multiplication (9) How many units are there in the ring Z Q Z (a) (b) (c) Infinitely many (d) none of these (9) The order of the ring M Z is : (a) (b) (c) ( d) None of these (9) A solution for the quadratic equation x x 0 in the ring Z is: (a) (b) (c) (d) (9). How many units are there in the ring Z Z (a) (b) (c) (9) Which of the following is not true (a) Every element in a ring has an additive inverse ( b) Multiplication in a field is commutative (c) Addition in every ring is commutative (d) Every ring with unity has at most two units (9) A solution for the quadratic equation x x 0 in the ring Z (a) - (b) (c) (d) is: (9) Which of the following ring have a non zero characteristic ( a) Z Z (b) Z Z (c) Z ( d) None of these Abstract Algebra

12 (9) What is the characteristic of Z Z (a) (b) (c) (99) Which of the following is true (a) The characteristic of nz is n. (b) nz has zero divisors if n is not a prime (c) As a ring, Z is isomorphic to nz; n (00) The solution for the equation x = in the field Z is: (a) (b) (c) (d) (0) The characteristic of the ring Z Z is: (a) (b) 0 (c) 90 (0) Let V be a vector space of dimension n. Which of the following is true (a) Any set containing n vectors is a basis for V (b) Any set containing n vectors linearly independent in V (c) Any set containing n vectors spans V (0) Which of the following sets of vectors form a basis of () {(0,,),(,0,),(0,,)} (){(,0,0),(0,0,),(0,,0)} (){(-,,),(,-,),(,-,)} (a) (b) (c) (d)all (0) Which of the following sets are vector spaces () Set of polynomials over R () Set of continuous functions on R ()Set of differentiable functions on R a)only & (b)only (c)all (d)none (0) The coordinate of (,,) relative to the ordered basis {(,0,)(,,0)(0,,)} of is (a) (,,) (b) (,,0) (c) (,,) (d)none (0) Let V be a vector space of all polynomials of degree n. Then dimension of V is (a)n (b)n- (c)n+ (d) (0) Let V be a vector space of dimension n. Which are true () Set of n+ vectors are linearly dependent ()V has only one basis ()Any set of n vectors is linearly independent (a)only (b) Only (c)only (d)all Abstract Algebra

13 (0) What is the dimension of the vector space of all matrices over R (a) (b) (c) (d) (b) (09) Which of the following is a subspace of (a) (, y, z) x y (c) (, y, z) x y z 0 R x (b) ( x, y, z) sin x 0 x (d) ( x, y, z) x y (0) Which of the following are true () is a ring () is a commutative ring () is a field () is a vector space over (a) & (b) Only (c) Only (d)all () How many ring homomorphisms are there from Z to Z (a) (b) (c) (d) () What is the dimension of the subspace W of, where W = {(a,b,c) : a=b=c } (a) (b) (c) (d) () Which of the following is a basis of the subspace W of, where W = {(a,b,c) : a=b=c } (a) {(,,)} (b) {(,,),(,,)} (c) {(,,)} () What is the co-ordinate vector of with respect to the ordered basis {, x, } (a) (0,,) (b)(0,0,) (c) (0,,0) (d)(,,) () Let W and W be subspaces of a vector space V. Then the smallest subspace of V containing both W and W is: (a) W (b). W (c) W W W W () Pick the incorrect statement: (a) If S spans V and S T, then T spans V (b) If T is linearly independent in V and S T, then S is linearly independent. (c) Any set of vectors that includes the zero vector is linearly dependent (d) Any single vector is linearly independent () Which of the following is a zero divisor in the ring (a) (b) (c) (d) 9 Abstract Algebra

14 () Value of m such that (m,, -) is a linear combination of the vectors (-,,) & (,,-) (a) (b) (c) (d) (9) The dimension of the subspace U = {(x,y,z) : x-y+z = 0} of is: (a) (b) (c) (b) (0) The coordinates of relative to the ordered basis (a) (,, ) (b) (,, ) (c) (,, ) (d) (,, ) Abstract Algebra

15 ANSWER KEY. b. c. a. d.c. c.b.b 9.a 0. b.b.b. c. b.d.b.a.b 9.a 0.c.b.d.d.c.b.a. d.b 9.a 0.b.c.d.a.d.b.a.a.b 9.a 0.a.b.d.b.a.c.a.d.d 9.b 0.a.b.d.a.d.c.a.c.a 9.a 0.a. b.d.d.b.a.a.b.b 9.a 0.d.a.c.c.d.d.a.c.c 9.b 0.a.c.b.a.c.c.b.a.b 9.d 90.d 9.c 9.c 9.d 9.b 9.d 9.c 9.d 9.a 99.d 00.c 0.b 0.d 0.b 0.c 0.a 0.c 0.a 0.b 09.c 0.a.b.a.a.a.c.d.b.b 9.b 0.a Abstract Algebra

### A Little Beyond: Linear Algebra

A Little Beyond: Linear Algebra Akshay Tiwary March 6, 2016 Any suggestions, questions and remarks are welcome! 1 A little extra Linear Algebra 1. Show that any set of non-zero polynomials in [x], no two

### 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

### 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

### GRE Subject test preparation Spring 2016 Topic: Abstract Algebra, Linear Algebra, Number Theory.

GRE Subject test preparation Spring 2016 Topic: Abstract Algebra, Linear Algebra, Number Theory. Linear Algebra Standard matrix manipulation to compute the kernel, intersection of subspaces, column spaces,

### 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

### Modern Computer Algebra

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

### 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

### INTRODUCTION TO THE GROUP THEORY

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

### 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.

### 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

### 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

### 5 Group theory. 5.1 Binary operations

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

### 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.

### IIT Mumbai 2015 MA 419, Basic Algebra Tutorial Sheet-1

IIT Mumbai 2015 MA 419, Basic Algebra Tutorial Sheet-1 Let Σ be the set of all symmetries of the plane Π. 1. Give examples of s, t Σ such that st ts. 2. If s, t Σ agree on three non-collinear points, then

### 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

### 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

### Rings and Fields Theorems

Rings and Fields Theorems Rajesh Kumar PMATH 334 Intro to Rings and Fields Fall 2009 October 25, 2009 12 Rings and Fields 12.1 Definition Groups and Abelian Groups Let R be a non-empty set. Let + and (multiplication)

### 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

### 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.

### 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

### 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

### 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:

### Algebra Exercises in group theory

Algebra 3 2010 Exercises in group theory February 2010 Exercise 1*: Discuss the Exercises in the sections 1.1-1.3 in Chapter I of the notes. Exercise 2: Show that an infinite group G has to contain a non-trivial

### 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

### 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

### 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

### Page Points Possible Points. Total 200

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

### CSIR - Algebra Problems

CSIR - Algebra Problems N. Annamalai DST - INSPIRE Fellow (SRF) Department of Mathematics Bharathidasan University Tiruchirappalli -620024 E-mail: algebra.annamalai@gmail.com Website: https://annamalaimaths.wordpress.com

### Some practice problems for midterm 2

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

### Mathematics for Cryptography

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

### Algebra Exam Topics. Updated August 2017

Algebra Exam Topics Updated August 2017 Starting Fall 2017, the Masters Algebra Exam will have 14 questions. Of these students will answer the first 8 questions from Topics 1, 2, and 3. They then have

### 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

### ERRATA. Abstract Algebra, Third Edition by D. Dummit and R. Foote (most recently revised on March 4, 2009)

ERRATA Abstract Algebra, Third Edition by D. Dummit and R. Foote (most recently revised on March 4, 2009) These are errata for the Third Edition of the book. Errata from previous editions have been fixed

### 18.312: Algebraic Combinatorics Lionel Levine. Lecture 22. Smith normal form of an integer matrix (linear algebra over Z).

18.312: Algebraic Combinatorics Lionel Levine Lecture date: May 3, 2011 Lecture 22 Notes by: Lou Odette This lecture: Smith normal form of an integer matrix (linear algebra over Z). 1 Review of Abelian

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

Abstract Algebra FINAL EXAM May 23, 2003 Name: R. Hammack Score: Directions: Please answer the questions in the space provided. To get full credit you must show all of your work. Use of calculators and

### Math 3121, A Summary of Sections 0,1,2,4,5,6,7,8,9

Math 3121, A Summary of Sections 0,1,2,4,5,6,7,8,9 Section 0. Sets and Relations Subset of a set, B A, B A (Definition 0.1). Cartesian product of sets A B ( Defintion 0.4). Relation (Defintion 0.7). Function,

### 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

### 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

### 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

### 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

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

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

### Moreover this binary operation satisfies the following properties

Contents 1 Algebraic structures 1 1.1 Group........................................... 1 1.1.1 Definitions and examples............................. 1 1.1.2 Subgroup.....................................

### 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

### 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

### 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,*)

### 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

### School of Mathematics and Statistics. MT5836 Galois Theory. Handout 0: Course Information

MRQ 2017 School of Mathematics and Statistics MT5836 Galois Theory Handout 0: Course Information Lecturer: Martyn Quick, Room 326. Prerequisite: MT3505 (or MT4517) Rings & Fields Lectures: Tutorials: Mon

### 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

### MATH HL OPTION - REVISION SETS, RELATIONS AND GROUPS Compiled by: Christos Nikolaidis

MATH HL OPTION - REVISION SETS, RELATIONS AND GROUPS Compiled by: Christos Nikolaidis PART B: GROUPS GROUPS 1. ab The binary operation a * b is defined by a * b = a+ b +. (a) Prove that * is associative.

### 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

### 12 Algebraic Structure and Coding Theory

12.1 The Structure of Algebra Def 1 Let S be a set and be a binary operation on S( : S S S). 2. The operation is associative over S, if a (b c) = (a b) c. 1. The operation is commutative over S, if a b

### 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

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

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

### 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

### 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

### Module MA3411: Abstract Algebra Galois Theory Michaelmas Term 2013

Module MA3411: Abstract Algebra Galois Theory Michaelmas Term 2013 D. R. Wilkins Copyright c David R. Wilkins 1997 2013 Contents 1 Basic Principles of Group Theory 1 1.1 Groups...............................

### 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

### 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

### NORTHERN INDIA ENGINEERING COLLEGE, LKO D E P A R T M E N T O F M A T H E M A T I C S. B.TECH IIIrd SEMESTER QUESTION BANK ACADEMIC SESSION

NORTHERN INDIA ENGINEERING COLLEGE, LKO D E P A R T M E N T O F M A T H E M A T I C S B.TECH IIIrd SEMESTER QUESTION BANK ACADEMIC SESSION 011-1 DISCRETE MATHEMATICS (EOE 038) 1. UNIT I (SET, RELATION,

### 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 Fields and vector spaces

1 Fields and vector spaces In this section we revise some algebraic preliminaries and establish notation. 1.1 Division rings and fields A division ring, or skew field, is a structure F with two binary

### Representation Theory

Part II Year 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2018 Paper 1, Section II 19I 93 (a) Define the derived subgroup, G, of a finite group G. Show that if χ is a linear character

### Math 546, Exam 2 Information.

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

### Course of algebra. Introduction and Problems

Course of algebra. Introduction and Problems A.A.Kirillov Fall 2008 1 Introduction 1.1 What is Algebra? Answer: Study of algebraic operations. Algebraic operation: a map M M M. Examples: 1. Addition +,

### 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

### 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)

### SCHOOL OF DISTANCE EDUCATION

SCHOOL OF DISTANCE EDUCATION B. Sc. MATHEMATICS MM5B06: ABSTRACT ALGEBRA (Core Course) FIFTH SEMESTER STUDY NOTES Prepared by: Vinod Kumar P. Assistant Professor P. G.Department of Mathematics T. M. Government

### UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION CORE COURSE. B.Sc. MATHEMATICS V SEMESTER. (2011 Admission onwards) BASIC MATHEMATICAL ANALYSIS

UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION CORE COURSE B.Sc. MATHEMATICS V SEMESTER (2011 Admission onwards) BASIC MATHEMATICAL ANALYSIS QUESTION BANK 1. Find the number of elements in the power

### Rings. Chapter 1. Definition 1.2. A commutative ring R is a ring in which multiplication is commutative. That is, ab = ba for all a, b R.

Chapter 1 Rings We have spent the term studying groups. A group is a set with a binary operation that satisfies certain properties. But many algebraic structures such as R, Z, and Z n come with two binary

### 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

### DEPARTMENT OF MATHEMATIC EDUCATION MATHEMATIC AND NATURAL SCIENCE FACULTY

HANDOUT ABSTRACT ALGEBRA MUSTHOFA DEPARTMENT OF MATHEMATIC EDUCATION MATHEMATIC AND NATURAL SCIENCE FACULTY 2012 BINARY OPERATION We are all familiar with addition and multiplication of two numbers. Both

### 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 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

### January 2016 Qualifying Examination

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

### Notation. 0,1,2,, 1 with addition and multiplication modulo

Notation Q,, The set of all natural numbers 1,2,3, The set of all integers The set of all rational numbers The set of all real numbers The group of permutations of distinct symbols 0,1,2,,1 with addition

### 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)

### ABSTRACT ALGEBRA MODULUS SPRING 2006 by Jutta Hausen, University of Houston

ABSTRACT ALGEBRA MODULUS SPRING 2006 by Jutta Hausen, University of Houston Undergraduate abstract algebra is usually focused on three topics: Group Theory, Ring Theory, and Field Theory. Of the myriad

### 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.

### Algebra Exam Syllabus

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

### Exercises MAT2200 spring 2013 Ark 3 Cosets, Direct products and Abelian groups

Exercises MAT2200 spring 2013 Ark 3 Cosets, Direct products and Abelian groups This Ark concerns the weeks No. (Feb ) andno. (Feb Mar ). The plans for those two weeks are as follows: On Wednesday Feb I

### 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

### 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 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

### TEST CODE: PMB SYLLABUS

TEST CODE: PMB SYLLABUS Convergence and divergence of sequence and series; Cauchy sequence and completeness; Bolzano-Weierstrass theorem; continuity, uniform continuity, differentiability; directional

### Part IV. Rings and Fields

IV.18 Rings and Fields 1 Part IV. Rings and Fields Section IV.18. Rings and Fields Note. Roughly put, modern algebra deals with three types of structures: groups, rings, and fields. In this section we

### Algebra Qualifying Exam Solutions. Thomas Goller

Algebra Qualifying Exam Solutions Thomas Goller September 4, 2 Contents Spring 2 2 2 Fall 2 8 3 Spring 2 3 4 Fall 29 7 5 Spring 29 2 6 Fall 28 25 Chapter Spring 2. The claim as stated is false. The identity

### 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

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

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

### 12 Algebraic Structure and Coding Theory

12.1 The Structure of Algebra Def 1 Let S be a set and be a binary operation on S( : S S S). 2. The operation is associative over S, if a (b c) = (a b) c. 1. The operation is commutative over S, if a b

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

### Section 18 Rings and fields

Section 18 Rings and fields Instructor: Yifan Yang Spring 2007 Motivation Many sets in mathematics have two binary operations (and thus two algebraic structures) For example, the sets Z, Q, R, M n (R)

### 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

### 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

### MATH 3005 ABSTRACT ALGEBRA I FINAL SOLUTION

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

### 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

### Mathematics 222a Quiz 2 CODE 111 November 21, 2002

Student s Name [print] Student Number Mathematics 222a Instructions: Print your name and student number at the top of this question sheet. Print your name and your instructor s name on the answer sheet.

### 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

### Homework 5 M 373K Mark Lindberg and Travis Schedler

Homework 5 M 373K Mark Lindberg and Travis Schedler 1. Artin, Chapter 3, Exercise.1. Prove that the numbers of the form a + b, where a and b are rational numbers, form a subfield of C. Let F be the numbers