# Supplement. Dr. Bob s Modern Algebra Glossary Based on Fraleigh s A First Course on Abstract Algebra, 7th Edition, Sections 0 through IV.

Size: px
Start display at page:

Download "Supplement. Dr. Bob s Modern Algebra Glossary Based on Fraleigh s A First Course on Abstract Algebra, 7th Edition, Sections 0 through IV."

Transcription

1 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 abelian if its binary operation is commutative. Alternating Group. The subgroup of S n consisting of the even permutations of n letters is the alternating group A n on n letters. Associative. A binary operation on a set S is associative if (a b) c = a (b c) for all a, b, c S. Automorphism of a Group. An isomorphism φ : G G of a group with itself is an automorphism of G. Binary Algebraic Structure. A binary algebraic structure is an ordered pair S, where S is a set and is a binary operation on S. Binary Operation. A binary operation on a set S is a function mapping S S into S. For each (ordered pair) (a, b) S S, we denote the element ((a, b)) S as a b. Cartesian Product. Let A and B be sets. The set A B = {(a, b) a A, b B} is the Cartesian product of A and B. The Cartesian product of sets S 1, S 2,..., S n of the set of all ordered n-tuples (a 1, a 2,..., a n ) where a i S i for i = 1, 2,..., n. This is denoted n S i = S 1 S 2 S n. i=1 Cayley Digraph/Graph. For a group G with generating set {a 1, a 2,..., a n }, define a digraph with vertex set V with the same elements as the elements of G. For each pair of vertices v 1 and v 2 define an arc (v 1, v 2 ) of color a i if v 1 a i = v 2. The totality of all arcs form the arc set A of the digraph. The vertex set V and arc set A together form a Cayley digraph for group G with respect to generating set {a 1, a 2,..., a n }. Center of a Group. For group G, define the center of G as Z(G) = {z g zg = gz for all g G}. Characteristic of a Ring. If for a ring R there is n N such that n a = 0 for all a R (remember that n a represents repeated addition), then the least such natural number is the characteristic of the ring R. If no such n exists, then ring R is of characteristic 0. Closed. Let be a binary operation on set S and let H S. Then H is closed under if for all a, b H, we also have a b H.

2 Glossary 2 Commutator Subgroup. For group G, consider the set C is the commutator subgroup of G. C = {aba 1 b 1 a, b G}. Commutative Binary Operation. A binary operation on a set S is commutative if a b = b a for all a, b S. Commutative Ring. A ring in which multiplication is commutative (i.e., ab = ba for all a, b R) is a commutative ring. Complex Numbers. The complex numbers are C = {a + ib a, b R, i 1}. Cosets. Let H be a subgroup of a group G. The subset ah = {ah h H} of G is the left coset of H containing a. The subset Ha = {ha h H} is the right coset of H containing a. Cycle. A permutation σ S n is a cycle if it has at most one orbit containing more than one element. The length of the cycle is the number of elements in its largest orbit. Cyclic Notation. Let σ S n be a cycle of length m where 1 < m n. Then the cyclic notation for σ is (a, σ(a), σ 2 (a),...σ m 1 (a)) where a is any element in the orbit of length m which results when {1, 2,..., n} is partitioned into orbits by σ. Cyclic Subgroup Generated by an Element. Let G be a group and let a G. Then the subgroup H = {a n n Z} of G (of Theorem 5.17) is the cyclic subgroup of G generated by a, denoted a. Cyclotomic Polynomial. The polynomial for prime p is the pth cyclotomic polynomial. Φ p (x) = xp 1 x 1 = xp 1 + x p x 2 + x + 1 Decomposable Group. A group G is decomposable if it is isomorphic to a direct product of two proper nontrivial subgroups. Otherwise G is indecomposable. Dihedral Group. The nth dihedral group D n is the group of symmetries of a regular n-gon. In fact, D n = 2n. Division Ring. Let R be a ring with unity 1 0. An element u R is a unit of R if it has a multiplicative inverse in R. If every nonzero element of R is a unit, then R is a division ring (or skew field). Divisors of Zero. If a and b are two nonzero elements of a ring R such that ab = 0 then a and b are divisors of 0.

3 Glossary 3 Equivalence Relation. An equivalence relation R on a set S is a relation on S such that for all x, y, z S, we have (1) R is reflexive: x Rx, (2) R is symmetric: If x Ry then y Rx, and (3) R is transitive: If x Ry and y Rz, then x Rx. Euler Phi-Function. For n N, define φ(n) as the number of natural numbers less than or equal to n which are relatively prime to n. φ is the Euler phi-function. Even and Odd Permutations. A permutation of a finite set is even or odd according to whether it can be expressed as a product of an even number of transpositions or the product of an odd number of transpositions. Factor Group (Quotient Group). Let H be a normal subgroup of G. Then the cosets of H form a group G/H under the binary operation (ah) (bh) = (ab)h called the the factor group (or quotient group) of G by H. Field. A field is a commutative division ring. Function. A function φ mapping set X into set Y is a relation between X and Y such that each x X appears as the first member of exactly one ordered pair (x, y) φ. We write φ : X Y and for (x, y) φ we write φ(x) = y. The domain of φ is the set X and the codomain of φ is Y. The range of φ is the set φ[x] = {φ(x) x X}. Generator of a Group. An element a of a group G generates G if a = G. A group is cyclic if there is a G such that a = G. Generating Set of a Group. Let G be a group and let a i G for i I. The smallest of G containing {a i i I} is the subgroup generated by the set {a i i I}. This subgroup is defined as the intersection of all subgroups of G containing {a i i I}: H = i J H j where the set of all subgroups of G containing {a i i I} is {H j j J}. If this subgroup is all of G, then the set {a i i I} generates G and the a i are generators of G. If there is a finite set {a i i I} that generates G, then G is finitely generated. Greatest Common Divisor. Let r, s N. The positive generator d of the cyclic group H = {nr + ms n, m Z} under addition is the greatest common divisor of r and s, denoted gcd(r, s).

4 Glossary 4 Group. A group G, is a set G and a binary operation on G such that G is closed under and G 1 For all a, b, c G, is associative: (a b) c = a (b c). G 2 There is e G called the identity such that for all x G: e x = x e = x. G 3 For all a G, there is an inverse a G such that: a a = a a = e. Homomorphism of Groups. A map φ of a group G into a group G is a homomorphism if for all a, b G we have φ(ab) = φ(a)φ(b). Homomorphism of Rings. For rings R and R, a map φ : R R is a homomorphism if for all a, b R we have: 1. φ(a + b) = φ(a) + φ(b), and 2. φ(ab) = φ(a)φ(b). Identity Element. Let S, be a binary structure. An element e of S is an identity element of if e s = s e = e for all s S. Image. Let f : A B for sets A and B. Let H A. The image of set H under f is {f(h) h H}, denoted f[h]. The inverse image of B in A is f 1 [B] = {a A f(a) B}. Index of a Subgroup. Let H be a subgroup of group G. The number of left cosets of H in G (technically, the cardinality of the set of left cosets) is the index of H in G, denoted (G : H). Integers. The integers are Z = {..., 3, 2, 1, 0, 1, 2, 3,...}. Integral Domain. An integral domain D is a commutative ring with unity 1 0 and containing no divisors of 0. Isomorphism of Binary Algebraic Structures. Let S, and S, be binary algebraic structures. An isomorphism of S with S is a one-to-one function φ mapping S onto S such that φ(x y) = φ(x) φ(y) for all x, y S. We then say S and S are isomorphic binary structures, denoted S S. Isomorphism of Rings. A isomorphism φ : R R from ring R to ring R is a homomorphism which is one to one and onto R. Kernel of a Homomorphism. Let φ : G G be a homomorphism. The subgroup φ 1 ({e }) = {x G φ(x) = e } (where e is the identity in G) is the kernel of φ, denoted Ker(φ).

5 Glossary 5 Least Common Multiple. For r 1, r 2,..., r n N, the smallest element of N that is a multiple of each r i for i = 1, 2,..., n, is the least common multiple of the r i, denoted lcm(r 1, r 2,..., r n ). Maximal Normal Subgroup. A maximal normal subgroup of a group G is a normal subgroup M not equal to G such that there is no proper normal subgroup N of G properly containing M. Modulus. For z = a + bi C, define the modulus or absolute value of z as z = a 2 + b 2. Natural Numbers. The natural numbers are N = {1, 2, 3,...}. Normal Subgroup. A subgroup H of a group G is normal if its left and right cosets coincide. That is if gh = Hg for all g G. Fraleigh simply says H is a normal subgroup of G, but a common notation is H G. Order of an Element. Let G be a group and a G. If G is cyclic and G = a, then (1) if G is finite of order n, then element a is of order n, and (2) if G is infinite then element a is of infinite order. Order of a Group. If G is a group, then the order G of G is the number of elements in G. One to One. A function φ : X Y is one to one (or an injection) if φ(x 1 ) = φ(x 2 ) implies x 1 = x 2. One-to-One Correspondence. A function that is both one to one and onto is called a one-to-one correspondence (or a bijection) between the domain and codomain. Onto. The function φ is onto Y (or a surjection) if the range of φ is Y. Partition. A partition of a set S is a collection of nonempty subsets of S such that every element of S is in exactly one of the subsets. Permutation. A permutation of a set A is a function φ : A A that is both one-to-one and onto. Polynomial over a Ring. Let R be a ring. A polynomial f(x) with coefficients in R is an infinite formal series a i x i = a 0 + a 1 x + a 2 x a n x n + i=0 where a i R and a i = 0 for all but a finite number of values of i. The a i are coefficients of f(x). If for some i 0 it is true that a i 0, then the largest such value of i is the degree of f(x). If all a i = 0, then the degree of f(x) is undefined. If a i = 0 for all i N, then f(x) is called a constant polynomial. We denote the set of all polynomials with coefficients in R as R[x]. Rational Numbers. The rational numbers are Q = {p/q p, q Z, q 0}. Real Numbers. The real numbers, denoted R, form a complete ordered field. Relation. A relation between sets A and B is a subset R of A B. if (a, b) R we say a is related to b, denoted a Rb.

6 Glossary 6 Ring. A ring R, +, is a set R together with two binary operations + and, called addition and multiplication, respectively, defined on R such that: R 1 : R, + is an abelian group. R 2 : Multiplication is associative: (a b) c = a (b c) for all a, b, c R. R 3 : For all a, b, c R, the left distribution law a (b + c) = (a c) + (b c) and the right distribution law (a + b) c = (a c) + (b c) hold. Ring with Unity. A ring with a multiplicative identity element is a ring with unity. The multiplicative unit is called unity. Same Cardinality. Two sets X and Y have the same cardinality if there exists a one to one function mapping X onto Y (that is, if there is a one-to-one correspondence between X and Y ). Simple Group. A group is simple if it is nontrivial and has no proper nontrivial normal subgroups. Strictly Skew Field. A noncommutative division ring is called a strictly skew field. Structural Property. A structural property of a binary structure S, is a property shared by any binary structure S, which is isomorphic to S,. Subgroup. If a subset H of a group G is closed under the binary operation of G and if H with the induced operation from G is itself a group, then H is a subgroup of G. We denote this as H G or G H. If H is a subgroup of G and H G, we write H < G or G > H. If G is a group, then G itself is a subgroup of G called the improper subgroup of G; all other subgroups are proper subgroups. The subgroup {e} is the trivial subgroup; all other subgroups are nontrivial subgroups. Symmetry Group. Let A be the finite set {1, 2,..., n}. The group of all permutations of A is the symmetric group on n letters, denoted S n. Transposition. A cycle of length 2 is a transposition. Whole Numbers. The whole numbers are W = {0, 1, 2, 3,...}. Zero of a Polynomial. Let F be a subfield of a field E, and let α E. Let f(x) = a 0 + a 1 x + a 2 x a n x n F[x] and let φ α : F[x] E be the evaluation homomorphism (see Theorem 22.4). We denote φ α (f(x)) = a 0 + a 1 α + a 2 α a n α n as f(α). If f(α) = 0, then α is a zero of f(x). Revised: 1/7/2013

### 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

### 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.

### 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.

### 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

### 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.

### 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

### 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

### 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

### RINGS: SUMMARY OF MATERIAL

RINGS: SUMMARY OF MATERIAL BRIAN OSSERMAN This is a summary of terms used and main results proved in the subject of rings, from Chapters 11-13 of Artin. Definitions not included here may be considered

### 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

### 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)

### 1 Rings 1 RINGS 1. Theorem 1.1 (Substitution Principle). Let ϕ : R R be a ring homomorphism

1 RINGS 1 1 Rings Theorem 1.1 (Substitution Principle). Let ϕ : R R be a ring homomorphism (a) Given an element α R there is a unique homomorphism Φ : R[x] R which agrees with the map ϕ on constant polynomials

### 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

### 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

### 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

### 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

### CS 468: Computational Topology Group Theory Fall b c b a b a c b a c b c c b a

Q: What s purple and commutes? A: An abelian grape! Anonymous Group Theory Last lecture, we learned about a combinatorial method for characterizing spaces: using simplicial complexes as triangulations

### 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

### 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

### 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

### Section 13 Homomorphisms

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

### B Sc MATHEMATICS ABSTRACT ALGEBRA

UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B Sc MATHEMATICS (0 Admission Onwards) V Semester Core Course ABSTRACT ALGEBRA QUESTION BANK () Which of the following defines a binary operation on Z

### 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

### 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)

### 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,

### 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

### Part II Permutations, Cosets and Direct Product

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

### 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.

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

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

### 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)

### Section 0. Sets and Relations

0. Sets and Relations 1 Section 0. Sets and Relations NOTE. Mathematics is the study of ideas, not of numbers!!! The idea from modern algebra which is the focus of most of this class is that of a group

### 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

### Factorization in Polynomial Rings

Factorization in Polynomial Rings Throughout these notes, F denotes a field. 1 Long division with remainder We begin with some basic definitions. Definition 1.1. Let f, g F [x]. We say that f divides g,

### 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

### 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

### 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

### Exercises MAT2200 spring 2014 Ark 4 Homomorphisms and factor groups

Exercises MAT2200 spring 2014 Ark 4 Homomorphisms and factor groups This Ark concerns the weeks No. (Mar ) and No. (Mar ). It is not very logical to have lectures on Fridays and problem solving in plenum

### Ideals, congruence modulo ideal, factor rings

Ideals, congruence modulo ideal, factor rings Sergei Silvestrov Spring term 2011, Lecture 6 Contents of the lecture Homomorphisms of rings Ideals Factor rings Typeset by FoilTEX Congruence in F[x] and

### Lecture 4.1: Homomorphisms and isomorphisms

Lecture 4.: Homomorphisms and isomorphisms Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4, Modern Algebra M. Macauley (Clemson) Lecture

### Finite Fields. Sophie Huczynska. Semester 2, Academic Year

Finite Fields Sophie Huczynska Semester 2, Academic Year 2005-06 2 Chapter 1. Introduction Finite fields is a branch of mathematics which has come to the fore in the last 50 years due to its numerous applications,

### Abstract Algebra II. Randall R. Holmes Auburn University

Abstract Algebra II Randall R. Holmes Auburn University Copyright c 2008 by Randall R. Holmes Last revision: November 30, 2009 Contents 0 Introduction 2 1 Definition of ring and examples 3 1.1 Definition.............................

### Foundations of Cryptography

Foundations of Cryptography Ville Junnila viljun@utu.fi Department of Mathematics and Statistics University of Turku 2015 Ville Junnila viljun@utu.fi Lecture 7 1 of 18 Cosets Definition 2.12 Let G be a

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

### 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

### Math Introduction to Modern Algebra

Math 343 - Introduction to Modern Algebra Notes Rings and Special Kinds of Rings Let R be a (nonempty) set. R is a ring if there are two binary operations + and such that (A) (R, +) is an abelian group.

### 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

### 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

### Finite Fields. Sophie Huczynska (with changes by Max Neunhöffer) Semester 2, Academic Year 2012/13

Finite Fields Sophie Huczynska (with changes by Max Neunhöffer) Semester 2, Academic Year 2012/13 Contents 1 Introduction 3 1 Group theory: a brief summary............................ 3 2 Rings and fields....................................

### Math 581 Problem Set 8 Solutions

Math 581 Problem Set 8 Solutions 1. Prove that a group G is abelian if and only if the function ϕ : G G given by ϕ(g) g 1 is a homomorphism of groups. In this case show that ϕ is an isomorphism. Proof:

### 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

### First Semester Abstract Algebra for Undergraduates

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

### 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

### 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

### Lecture Notes Math 371: Algebra (Fall 2006) by Nathanael Leedom Ackerman

Lecture Notes Math 371: Algebra (Fall 2006) by Nathanael Leedom Ackerman October 17, 2006 TALK SLOWLY AND WRITE NEATLY!! 1 0.1 Integral Domains and Fraction Fields 0.1.1 Theorems Now what we are going

### A Little Beyond: Linear Algebra

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

### Algebra Review. Instructor: Laszlo Babai Notes by Vincent Lucarelli and the instructor. June 15, 2001

Algebra Review Instructor: Laszlo Babai Notes by Vincent Lucarelli and the instructor June 15, 2001 1 Groups Definition 1.1 A semigroup (G, ) is a set G with a binary operation such that: Axiom 1 ( a,

### 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

### 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

### 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

### 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

### 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

### Math 547, Exam 1 Information.

Math 547, Exam 1 Information. 2/10/10, LC 303B, 10:10-11:00. Exam 1 will be based on: Sections 5.1, 5.2, 5.3, 9.1; The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/547sp10/547.html)

### 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

### 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

### 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,

### 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

### List of topics for the preliminary exam in algebra

List of topics for the preliminary exam in algebra 1 Basic concepts 1. Binary relations. Reflexive, symmetric/antisymmetryc, and transitive relations. Order and equivalence relations. Equivalence classes.

### 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 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

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

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

### 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

### 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

### MA441: Algebraic Structures I. Lecture 14

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

### Graduate Preliminary Examination

Graduate Preliminary Examination Algebra II 18.2.2005: 3 hours Problem 1. Prove or give a counter-example to the following statement: If M/L and L/K are algebraic extensions of fields, then M/K is algebraic.

### GALOIS THEORY. Contents

GALOIS THEORY MARIUS VAN DER PUT & JAAP TOP Contents 1. Basic definitions 1 1.1. Exercises 2 2. Solving polynomial equations 2 2.1. Exercises 4 3. Galois extensions and examples 4 3.1. Exercises. 6 4.

### 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 =

### 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

### Abstract Algebra II. Randall R. Holmes Auburn University. Copyright c 2008 by Randall R. Holmes Last revision: November 7, 2017

Abstract Algebra II Randall R. Holmes Auburn University Copyright c 2008 by Randall R. Holmes Last revision: November 7, 2017 This work is licensed under the Creative Commons Attribution- NonCommercial-NoDerivatives

### 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

### 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

### Reducibility of Polynomials over Finite Fields

Master Thesis Reducibility of Polynomials over Finite Fields Author: Muhammad Imran Date: 1976-06-02 Subject: Mathematics Level: Advance Course code: 5MA12E Abstract Reducibility of certain class of polynomials

### 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)

### 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

### SF2729 GROUPS AND RINGS LECTURE NOTES

SF2729 GROUPS AND RINGS LECTURE NOTES 2011-03-01 MATS BOIJ 6. THE SIXTH LECTURE - GROUP ACTIONS In the sixth lecture we study what happens when groups acts on sets. 1 Recall that we have already when looking

### 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.

### Section VI.33. Finite Fields

VI.33 Finite Fields 1 Section VI.33. Finite Fields Note. In this section, finite fields are completely classified. For every prime p and n N, there is exactly one (up to isomorphism) field of order p n,

### 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.

### Algebra Homework, Edition 2 9 September 2010

Algebra Homework, Edition 2 9 September 2010 Problem 6. (1) Let I and J be ideals of a commutative ring R with I + J = R. Prove that IJ = I J. (2) Let I, J, and K be ideals of a principal ideal domain.

### MATH 101: ALGEBRA I WORKSHEET, DAY #1. We review the prerequisites for the course in set theory and beginning a first pass on group. 1.

MATH 101: ALGEBRA I WORKSHEET, DAY #1 We review the prerequisites for the course in set theory and beginning a first pass on group theory. Fill in the blanks as we go along. 1. Sets A set is a collection

### Chapter 0. Introduction: Prerequisites and Preliminaries

Chapter 0. Sections 0.1 to 0.5 1 Chapter 0. Introduction: Prerequisites and Preliminaries Note. The content of Sections 0.1 through 0.6 should be very familiar to you. However, in order to keep these notes

### 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

### 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

### Algebra Prelim Notes

Algebra Prelim Notes Eric Staron Summer 2007 1 Groups Define C G (A) = {g G gag 1 = a for all a A} to be the centralizer of A in G. In particular, this is the subset of G which commuted with every element

### 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

### 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

### 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

### 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 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