Examples: The (left or right) cosets of the subgroup H = 11 in U(30) = {1, 7, 11, 13, 17, 19, 23, 29} are
|
|
- Dwight Moore
- 6 years ago
- Views:
Transcription
1 Cosets Let H be a subset of the group G. (Usually, H is chosen to be a subgroup of G.) If a G, then we denote by ah the subset {ah h H}, the left coset of H containing a. Similarly, Ha = {ha h H} is the right coset of H containing a. In both situations, we call a the coset representative. Examples: The (left or right) cosets of the subgroup H = 11 in U(30) = {1, 7, 11, 13, 17, 19, 23, 29} are 1H = {1, 11} = 11H, 7H = {7, 17} = 17H, 13H = {13, 23} = 23H, 19H = {19, 29} = 29H. The (left or right) cosets of the subgroup 4Z in the additive group Z are 4Z = 0 + 4Z = ±4 + 4Z = ±8 + 4Z = ±12 + 4Z = L, L = 3 + 4Z = 1 + 4Z = 5 + 4Z = 9 + 4Z = L, L = 2 + 4Z = 2 + 4Z = 6 + 4Z = Z = L, L = 1 + 4Z = 3 + 4Z = 7 + 4Z = Z = L.
2 The left cosets of the subgroup {R 0,H} in D 4 are R 0 { R 0,H} = {R 0,H} = H{ R 0,H}, R 90 { R 0,H} = {R 90, D } = D {R 0,H}, R 180 { R 0,H} = {R 180,V} = V {R 0,H}, R 270 { R 0,H} = {R 270, D} = D{R 0,H}. There are a number of patterns evidenced in these examples that are true more generally. Theorem Let H be a subgroup of G and suppose a,b G. Then the left cosets of H in G satisfy the following properties (with correspondingly similar results for the right cosets of H): 1. a ah, 2. ah = H a H, 3. either ah = bh or ah bh =, 4. ah = bh a 1 b H, 5. ah = bh, 6. ah = Ha H = aha 1, 7. ah is a subgroup of G a H. Proof 1. Trivial. 2. ( ) a = ae ah = H. ( ) a H a 1 H and for every h H, ah H by closure in H, whence ah H ; and h = a(a 1 h) ah, whence ah H. Thus, ah = H.
3 3. Suppose ah bh. Then there is some x that lies in both cosets; that is, there are elements y,z H for which x = ay = bz. Thus, ah = (bzy 1 )H = b(zy 1 H ) = bh (by #2 above, since zy 1 H ). 4. ah = bh H = a 1 bh a 1 b H by #2 above. 5. The function from the set ah to the set bh given by ah a bh is clearly onto. It is one-to-one since bh 1 = bh 2 h 1 = h 2 ah 1 = ah 2. So ah = bh. 6. ah = Ha aha 1 = (ah )a 1 = (Ha)a 1 = He = H. 7. ah G e ah ah H ah = H a H (using #3 and #2 above). Conversely, by #2, a H ah = H G. // Properties #1, #3, and #5 above combine to assert that G is partitioned by the left cosets of H into subsets of the same size. In fact, we can define a relation on the elements of G by a ~ b when a and b lie in the same coset, i.e., ah = bh. This relation is reflexive (#1), symmetric, and transitive (by #3), so it is an equivalence relation on G. (See pp ) Examples: Let Π be a plane through the origin in R 3. Then Π is an additive subgroup of R 3. The cosets of Π represent all the planes in R 3 parallel to Π. The cosets of SL(2, R) partition GL(2, R) into subsets consisting of all matrices with a given determinant.
4 Lagrange s Theorem One of the most versatile results in all of group theory is Lagrange s Theorem Let G be a finite group and suppose that H < G. Then H divides G. Moreover, the number of distinct left (or right) cosets of H in G, called the index of H in G and denoted G:H, equals G / H. Proof As we mentioned in the comment following the previous theorem, properties #1, #3 and #5 imply that G is the disjoint union of the distinct cosets of H, all of which have the same number of elements. Since H is one of these cosets, it follows that G is the disjoint union of G:H cosets, all of which have size H. That is, G = G:H H, from which the final claim of the theorem follows. // Corollary The order of any element of a finite group divides the order of the group. Proof For any element a of a group, a = a. Corollary Groups of prime order must be cyclic. Proof Let a be an element of a group G of prime order p which is not the identity element. Then a divides p but a 1. So a = p and G = a. //
5 Corollary If G is a finite group and a G, then a G = e. Proof a must divide G. // Corollary [Fermat s Little Theorem (F lt)] If p is a prime number and a is any integer, then a p mod p = amod p. Proof If a is a multiple of p, then both amod p and a p mod p equal 0. If a is not a multiple of p, then both these numbers are elements of the multiplicative group U(p), which has order p 1. Therefore, a p 1 mod p = 1. Multiplication by a yields a p mod p = amod p. //
6 Lagrange s Theorem says that a group of order n can only have subgroups whose orders are divisors of n. Note however that the converse of this theorem need not be true: Proposition A 4 = 12, but A 4 has no subgroup of order 6. Proof Suppose H were a subgroup of order 6. Then since A 4 :H = 2, we can conclude that H has exactly two distinct cosets in A 4. Let α represent any one of the eight elements of order 3 in A 4 ( α 5,α 6,K,α 12 in the Cayley table of A 4 on p. 105). Then at least one pair of the three cosets H,αH,α 2 H must coincide and regardless which pair it is, we must then have that α H. But then H must contain all eight elements of order 3, which is impossible. //
7 Lagrange s Theorem gives us a powerful tool for investigating the nature of finite groups. For example: Theorem A group whose order is twice an odd prime p must be isomorphic to either Z 2 p or D p. Proof If p is an odd prime and G has order 2p, we have two cases to consider: (1) G has an element of order 2p. Then clearly G Z 2 p. (2) G has no element of order 2p. By Lagrange s Theorem, any element different from e must have order 2 or order p. If every element had order 2, then every element would be its own inverse and we could write xy = (xy) 1 = y 1 x 1 = yx, showing that G is Abelian; moreover, this would allow any two such elements to generate a subgroup {e,x, y,xy} of order 4. But 4 does not divide 2p, so this is not possible. Therefore, not every element has order 2 and there must be some element a of order p. Now let b be an element not in a. Then the coset b a is distinct from a, and since G: a = 2, these are the only two distinct cosets. So b 2 a must conicide with one of them. Since b 2 a = b a implies b a = a, which is false, we must have b 2 a = a b 2 a. So b 2 divides a = p. It
8 follows that b 2 = 1, for if b 2 = p, the fact that b 2p together with b 2 b <G means that b = p, and since b 2 a, b = b p+1 = (b 2 ) p +1 2 = (a k ) p +1 2 a, which we know to be false. Thus b 2 = 1, and b must have order 2. Next, ab a (since ab a b a ), so the same argument we used in the last paragraph will show that ab = 2. Also, ab = (ab) 1 = b 1 a 1 = ba 1. Therefore, G = {e,a,a 2,,a p 1,b,ba,ba 2,,ba p 1 } and the group operation is determined by the relations between a and b. In particular, a k b = a k 1 (ab) = a k 1 (ba 1 ) = a k 1 ba 1 = a k 2 (ab)a 1 = a k 2 (ba 1 )a 1 = a k 2 ba 2 =L = ba k so the product of any two elements in G is completely determined, as follows:
9 a k a l = a k+l ; (ba k )a l = ba k+l ; a k (ba l ) = ba l k ; (ba k )(ba l ) = a l k. In other words, every group of order 2p without an element of order 2p must have this structure: all such groups are isomorphic. As D p is a group of this form, all such groups are isomorphic to D p. // Corollary S 3 D 3. //
10 Orbits and Stabilizers We saw from Cayley s Theorem how every group is a group of permutations on some set. Suppose that G is a group of permutations of the objects in the set S. Then for each s S, the set orb G (s) = {ϕ(s) S ϕ G}, called the orbit of s under G, is the subset of elements in S to which s is moved under the action of G. We also define stab G (s) = {ϕ G ϕ(s) = s}, called the stabilizer of s in G, to be the subset of G consisting of those elements that fix s. As this (nonempty) subset of G is clearly closed under composition and under the taking of inverses, it is a subgroup of G.
11 Theorem Let G be a finite group of permutations of the set S. Then for any s S, G = orb G (s) stab G (s). Proof By Lagrange s Theorem, G / stab G (s) is the number of left cosets of stab G (s) in G. So we are done if we can establish a bijection between the left cosets of stab G (s) and the orbit of s under G. To this end, define the function f as ϕstab G (s) aϕ(s). Before we continue, we must first show that this is well-defined; that is, the coset ϕstab G (s) may be equal to the coset ψ stab G (s), so we must ensure that ϕ(s) = ψ (s). But this is always true, for ϕstab G (s) = ψ stab G (s) ϕ 1 ψ stab G (s), which means that ϕ 1 ψ (s) = s, or ϕ(s) = ψ (s). The function f is one-to-one because we can reverse this last argument, and it is onto because if t is in the orbit of s, then t =ϕ(s) for some ϕ G, whence f maps the coset ϕstab G (s) to t =ϕ(s). //
MA441: Algebraic Structures I. Lecture 18
MA441: Algebraic Structures I Lecture 18 5 November 2003 1 Review from Lecture 17: Theorem 6.5: Aut(Z/nZ) U(n) For every positive integer n, Aut(Z/nZ) is isomorphic to U(n). The proof used the map T :
More informationAnswers to Final Exam
Answers to Final Exam MA441: Algebraic Structures I 20 December 2003 1) Definitions (20 points) 1. Given a subgroup H G, define the quotient group G/H. (Describe the set and the group operation.) The quotient
More informationHomework 7 Solution Chapter 7 - Cosets and Lagrange s theorem. due: Oct. 31.
Homework 7 Solution Chapter 7 - Cosets and Lagrange s theorem. due: Oct. 31. 1. Find all left cosets of K in G. (a) G = Z, K = 4. K = {4m m Z}, 1 + K = {4m + 1 m Z}, 2 + K = {4m + 2 m Z}, 3 + K = {4m +
More informationDISCRETE 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
More informationDefinitions. 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
More information0 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
More informationQuiz 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.
More informationWe begin with some definitions which apply to sets in general, not just groups.
Chapter 8 Cosets In this chapter, we develop new tools which will allow us to extend to every finite group some of the results we already know for cyclic groups. More specifically, we will be able to generalize
More informationChapter 5 Groups of permutations (bijections) Basic notation and ideas We study the most general type of groups - groups of permutations
Chapter 5 Groups of permutations (bijections) Basic notation and ideas We study the most general type of groups - groups of permutations (bijections). Definition A bijection from a set A to itself is also
More informationMATH 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
More informationBasic 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
More informationDefinitions, 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
More informationStab(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
More informationHomework #11 Solutions
Homework #11 Solutions p 166, #18 We start by counting the elements in D m and D n, respectively, of order 2. If x D m and x 2 then either x is a flip or x is a rotation of order 2. The subgroup of rotations
More informationGroups 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
More information6 Cosets & Factor Groups
6 Cosets & Factor Groups The course becomes markedly more abstract at this point. Our primary goal is to break apart a group into subsets such that the set of subsets inherits a natural group structure.
More informationSupplement. 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
More informationINTRODUCTION 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
More informationMATH 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.
More informationDefinition 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.
More informationAlgebraic 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
More informationCosets. gh = {gh h H}. Hg = {hg h H}.
Cosets 10-4-2006 If H is a subgroup of a group G, a left coset of H in G is a subset of the form gh = {gh h H}. A right coset of H in G is a subset of the form Hg = {hg h H}. The collection of left cosets
More information1.1 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
More informationAlgebra: Groups. Group Theory a. Examples of Groups. groups. The inverse of a is simply a, which exists.
Group Theory a Let G be a set and be a binary operation on G. (G, ) is called a group if it satisfies the following. 1. For all a, b G, a b G (closure). 2. For all a, b, c G, a (b c) = (a b) c (associativity).
More informationSolutions 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
More information( ) 3 = ab 3 a!1. ( ) 3 = aba!1 a ( ) = 4 " 5 3 " 4 = ( )! 2 3 ( ) =! 5 4. Math 546 Problem Set 15
Math 546 Problem Set 15 1. Let G be a finite group. (a). Suppose that H is a subgroup of G and o(h) = 4. Suppose that K is a subgroup of G and o(k) = 5. What is H! K (and why)? Solution: H! K = {e} since
More informationFirst 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
More informationHomework #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
More informationMath 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
More information1 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
More informationCHAPTER 9. Normal Subgroups and Factor Groups. Normal Subgroups
Normal Subgroups CHAPTER 9 Normal Subgroups and Factor Groups If H apple G, we have seen situations where ah 6= Ha 8 a 2 G. Definition (Normal Subgroup). A subgroup H of a group G is a normal subgroup
More informationSF2729 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
More informationFROM GROUPS TO GALOIS Amin Witno
WON Series in Discrete Mathematics and Modern Algebra Volume 6 FROM GROUPS TO GALOIS Amin Witno These notes 1 have been prepared for the students at Philadelphia University (Jordan) who are taking the
More informationAlgebra 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
More information) = 1, ) = 2, and o( [ 11]
True/False Questions 1. The order of the identity element in any group is 1. True. n = 1 is the least positive integer such that e n = e. 2. Every cyclic group is abelian. True. Let G be a cyclic group.
More information7. 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
More informationSome 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
More informationGroups. 3.1 Definition of a Group. Introduction. Definition 3.1 Group
C H A P T E R t h r e E Groups Introduction Some of the standard topics in elementary group theory are treated in this chapter: subgroups, cyclic groups, isomorphisms, and homomorphisms. In the development
More informationModern 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
More informationCONSEQUENCES 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.
More informationWritten Homework # 2 Solution
Math 516 Fall 2006 Radford Written Homework # 2 Solution 10/09/06 Let G be a non-empty set with binary operation. For non-empty subsets S, T G we define the product of the sets S and T by If S = {s} is
More informationMath 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
More informationSection 10: Counting the Elements of a Finite Group
Section 10: Counting the Elements of a Finite Group Let G be a group and H a subgroup. Because the right cosets are the family of equivalence classes with respect to an equivalence relation on G, it follows
More information5 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
More informationMATH 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
More information17 More Groups, Lagrange s Theorem and Direct Products
7 More Groups, Lagrange s Theorem and Direct Products We consider several ways to produce groups. 7. The Dihedral Group The dihedral group D n is a nonabelian group. This is the set of symmetries of a
More informationMath 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)
More informationGroups 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,*)
More informationCosets and Normal Subgroups
Cosets and Normal Subgroups (Last Updated: November 3, 2017) These notes are derived primarily from Abstract Algebra, Theory and Applications by Thomas Judson (16ed). Most of this material is drawn from
More information15 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
More informationMath 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
More informationThe 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
More informationSection 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
More informationTheorems 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
More informationModern 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
More informationREMARKS 7.6: Let G be a finite group of order n. Then Lagrange's theorem shows that the order of every subgroup of G divides n; equivalently, if k is
FIRST-YEAR GROUP THEORY 7 LAGRANGE'S THEOREM EXAMPLE 7.1: Set G = D 3, where the elements of G are denoted as usual by e, a, a 2, b, ab, a 2 b. Let H be the cyclic subgroup of G generated by b; because
More informationGroup 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
More informationA. (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 =
More informationDEPARTMENT 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
More informationSolutions of Assignment 10 Basic Algebra I
Solutions of Assignment 10 Basic Algebra I November 25, 2004 Solution of the problem 1. Let a = m, bab 1 = n. Since (bab 1 ) m = (bab 1 )(bab 1 ) (bab 1 ) = ba m b 1 = b1b 1 = 1, we have n m. Conversely,
More information3.3 Equivalence Relations and Partitions on Groups
84 Chapter 3. Groups 3.3 Equivalence Relations and Partitions on Groups Definition 3.3.1. Let (G, ) be a group and let H be a subgroup of G. Let H be the relation on G defined by a H b if and only if ab
More informationCosets, 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
More informationLagrange s Theorem. Philippe B. Laval. Current Semester KSU. Philippe B. Laval (KSU) Lagrange s Theorem Current Semester 1 / 10
Lagrange s Theorem Philippe B. Laval KSU Current Semester Philippe B. Laval (KSU) Lagrange s Theorem Current Semester 1 / 10 Introduction In this chapter, we develop new tools which will allow us to extend
More informationLECTURES 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
More informationMath 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,
More informationMath 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:
More informationPart 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
More informationD-MATH Algebra I HS18 Prof. Rahul Pandharipande. Solution 1. Arithmetic, Zorn s Lemma.
D-MATH Algebra I HS18 Prof. Rahul Pandharipande Solution 1 Arithmetic, Zorn s Lemma. 1. (a) Using the Euclidean division, determine gcd(160, 399). (b) Find m 0, n 0 Z such that gcd(160, 399) = 160m 0 +
More informationBefore you begin read these instructions carefully.
MATHEMATICAL TRIPOS Part IA Tuesday, 4 June, 2013 1:30 pm to 4:30 pm PAPER 3 Before you begin read these instructions carefully. The examination paper is divided into two sections. Each question in Section
More informationMA441: 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
More informationMODEL ANSWERS TO THE FIFTH HOMEWORK
MODEL ANSWERS TO THE FIFTH HOMEWORK 1. Chapter 3, Section 5: 1 (a) Yes. Given a and b Z, φ(ab) = [ab] = [a][b] = φ(a)φ(b). This map is clearly surjective but not injective. Indeed the kernel is easily
More informationNormal 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
More informationThe number of ways to choose r elements (without replacement) from an n-element set is. = r r!(n r)!.
The first exam will be on Friday, September 23, 2011. The syllabus will be sections 0.1 through 0.4 and 0.6 in Nagpaul and Jain, and the corresponding parts of the number theory handout found on the class
More informationTeddy 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
More informationGROUPS. Chapter-1 EXAMPLES 1.1. INTRODUCTION 1.2. BINARY OPERATION
Chapter-1 GROUPS 1.1. INTRODUCTION The theory of groups arose from the theory of equations, during the nineteenth century. Originally, groups consisted only of transformations. The group of transformations
More informationGroups. 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
More information7 Semidirect product. Notes 7 Autumn Definition and properties
MTHM024/MTH74U Group Theory Notes 7 Autumn 20 7 Semidirect product 7. Definition and properties Let A be a normal subgroup of the group G. A complement for A in G is a subgroup H of G satisfying HA = G;
More informationFoundations 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
More information120A 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
More informationLecture 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:
More informationMath 31 Take-home Midterm Solutions
Math 31 Take-home Midterm Solutions Due July 26, 2013 Name: Instructions: You may use your textbook (Saracino), the reserve text (Gallian), your notes from class (including the online lecture notes), and
More informationSection 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
More informationDiscrete Mathematics with Applications MATH236
Discrete Mathematics with Applications MATH236 Dr. Hung P. Tong-Viet School of Mathematics, Statistics and Computer Science University of KwaZulu-Natal Pietermaritzburg Campus Semester 1, 2013 Tong-Viet
More information1 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
More informationIntroduction 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)
More informationMATH 433 Applied Algebra Lecture 22: Review for Exam 2.
MATH 433 Applied Algebra Lecture 22: Review for Exam 2. Topics for Exam 2 Permutations Cycles, transpositions Cycle decomposition of a permutation Order of a permutation Sign of a permutation Symmetric
More informationSupplementary Notes: Simple Groups and Composition Series
18.704 Supplementary Notes: Simple Groups and Composition Series Genevieve Hanlon and Rachel Lee February 23-25, 2005 Simple Groups Definition: A simple group is a group with no proper normal subgroup.
More information2 Lecture 2: Logical statements and proof by contradiction Lecture 10: More on Permutations, Group Homomorphisms 31
Contents 1 Lecture 1: Introduction 2 2 Lecture 2: Logical statements and proof by contradiction 7 3 Lecture 3: Induction and Well-Ordering Principle 11 4 Lecture 4: Definition of a Group and examples 15
More informationHW2 Solutions Problem 1: 2.22 Find the sign and inverse of the permutation shown in the book (and below).
Teddy Einstein Math 430 HW Solutions Problem 1:. Find the sign and inverse of the permutation shown in the book (and below). Proof. Its disjoint cycle decomposition is: (19)(8)(37)(46) which immediately
More informationMath 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
More informationMath 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
More informationA conjugacy criterion for Hall subgroups in finite groups
MSC2010 20D20, 20E34 A conjugacy criterion for Hall subgroups in finite groups E.P. Vdovin, D.O. Revin arxiv:1004.1245v1 [math.gr] 8 Apr 2010 October 31, 2018 Abstract A finite group G is said to satisfy
More informationIsomorphisms. 0 a 1, 1 a 3, 2 a 9, 3 a 7
Isomorphisms Consider the following Cayley tables for the groups Z 4, U(), R (= the group of symmetries of a nonsquare rhombus, consisting of four elements: the two rotations about the center, R 8, and
More informationSUBGROUPS OF CYCLIC GROUPS. 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by
SUBGROUPS OF CYCLIC GROUPS KEITH CONRAD 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by g = {g k : k Z}. If G = g, then G itself is cyclic, with g as a generator. Examples
More informationAlgebra I. Randall R. Holmes Auburn University
Algebra I Randall R. Holmes Auburn University Copyright c 2008 by Randall R. Holmes Last revision: June 8, 2009 Contents 0 Introduction 2 1 Definition of group and examples 4 1.1 Definition.............................
More informationGROUPS AS GRAPHS. W. B. Vasantha Kandasamy Florentin Smarandache
GROUPS AS GRAPHS W. B. Vasantha Kandasamy Florentin Smarandache 009 GROUPS AS GRAPHS W. B. Vasantha Kandasamy e-mail: vasanthakandasamy@gmail.com web: http://mat.iitm.ac.in/~wbv www.vasantha.in Florentin
More informationFinite Fields. Saravanan Vijayakumaran Department of Electrical Engineering Indian Institute of Technology Bombay
1 / 25 Finite Fields Saravanan Vijayakumaran sarva@ee.iitb.ac.in Department of Electrical Engineering Indian Institute of Technology Bombay September 25, 2014 2 / 25 Fields Definition A set F together
More informationDirection: 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,
More informationHomomorphisms. The kernel of the homomorphism ϕ:g G, denoted Ker(ϕ), is the set of elements in G that are mapped to the identity in G.
10. Homomorphisms 1 Homomorphisms Isomorphisms are important in the study of groups because, being bijections, they ensure that the domain and codomain groups are of the same order, and being operation-preserving,
More information2) 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
More information