INTRODUCTION TO THE GROUP THEORY


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2 Lecture Notes on Structure of Algebra INTRODUCTION TO THE GROUP THEORY By : Drs. Antonius Cahya Prihandoko, M.App.Sc Mathematics Education Study Program Faculty of Teacher Training and Education The University of Jember
3 Acknowledgments Thanks to the Lord, because of His Mercy, this book has been finished. This book is written as one of the resources of the subject of Algebra Structure. The discussion focus in this book are group and homomorphism. The description is started with the concepts of set and function, that are basic of all concepts in this book. After getting a good knowledge in the concepts of set and function, the students can continue to the chapter 2 that provides construction, properties and order of a group and its subgroup. The next discussion on group including cyclic group, the group of permutations and the Lagrange theorem, that will be available in the chapter 3, 4, dan 5, respectively. The second part of the discussion is about homomorphism. This second part is started with the chapter 6 that provides homomorphism, isomorphism and Cayley theorem. Finally, the last chapter provides the concept of factor group. Jember, August 2010 Antonius C. Prihandoko i
4 Table of Contents Acknowledgments Table of Contents i iii 1 SET AND FUNCTION Set Partition and Equivalence Relation Function Binary Operation Exercise on Set and Function GROUP Definition of Group Properties of Group Order of Group and Element Subgroup The Exercise for Concepts of Group CYCLIC GROUP Concept and Basic Properties Subgroup of Finite Cyclic Group Exercises on Cyclic Group GROUP OF PERMUTATIONS Permutation ii
5 4.2 Orbit and Cycle Exercises on Group of Permutation COSET AND THE LAGRANGE THEOREM Coset The Lagrange Theorem Exercises on Coset and the Lagrange Theorem GROUP HOMOMORPHISM Homomorphism Isomorphism dan Cayley Theorem Exercises on group Homomorphism THE FACTOR GROUP Factor Group Development by a Homomorphism Factor Group Development by Normal Subgroup The Fundamental Homomorphism Theorem Exercises on the Factor Group iii
6 Chapter 1 SET AND FUNCTION This chapter provides the materials needed to reach the main substances: (group and homomorphism), in the subject of Structure of Algebra. As a group is basically a set and a homomorphism is a function, it is important to describe set theory and function first before discussing the main contents of this subject. The aim of this chapter is that the students have understanding on set, function, partition, equivalence relation and binary operation. Outcome of this chapter is that the students are able to 1. solve the set operation; 2. show an equivalence relation; 3. show a partition of a set; 4. construct a function; 5. show a binary operation; 1.1 Set Not all concepts in mathematics can be well defined, sometime a concept can be understood by identifying its properties. The concept of set for instance, if set is identified as group of certain objects, then there will be a question about definition of group. If group is identified as unity of things, then there will be a question about definition of unity. This sequence of questions will be unstopped, or we will repeat the words in previous definition. Therefore, in this chapter, set will not be defined, but it will be identified by analyzing its characteristics. Briefly, several things related to set can be described below. 1
7 I. Set and Function antonius cp 2 1. A set S consists of elements, and if a is an element of S, then it can be notated as a S. 2. There is exactly one set that has no element. It is called empty set, dan be notated as φ. 3. A set can be described by identifying its properties, or by listing its elements. For example, set of prime numbers less than or equal to 5, can be described as {2, 3, 5}, or {x x primes 5}. 4. A set is called welldefined, if it can be decided definitely whether an object is element or not. Let S = {some natural numbers }, then S is not welldefined set because it can not be decided whether 5 S or 5 S. If S = {the first four natural numbers}, then elements of S can be definitely described, that is 1, 2, 3, 4. Definition A set B is subset of set A and be notated B A or A B, if every element of B is also element of A. Note : For every set A, Both of A and φ are subset of A. A is improper subset, while the others are proper subset. Example Let S = {a, b, c}, then S has 8 subsets that is φ, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}. The concept of subset can be used to prove the equality of two sets, i.e. two sets A and B are same if A B and B A. 1.2 Partition and Equivalence Relation Definition A partition of set A is a family set consisting of disjoint non empty subsets which is union of them constructs the set A Example {{a, b}, {c}} is a partition of set S = {a, b, c}.
8 I. Set and Function antonius cp 3 Based on that definition, proving that a family set {A 1, A 2, A 3,..., A n } is a partition of set A, can be shown below : 1. i, j {1, 2, 3,..., n}, if i j then A i A j = φ; 2. n i=1a i = A Example The integers set, Z, can be partitioned to set of odd integers and set of even integers. 2. Z can be partitioned into classes of residues. Another concept closely related to the partition is equivalence relation. If a set is partitioned then there is an equivalence relation that can be found in that set. Vice versa, if an equivalence relation is applied to a set, then the set of all equivalence classes forms a partition on the set. Definition A relation on a set A is an equivalence relation if and only if it is: 1. reflexive; i.e x A, x x; 2. symmetric; i.e if x y then y x; 3. transitive; i.e if x y and y z then x z. Example relation. 1. Relation same as in the real set, R, is an equivalence 2. Let on the rational numbers set, Q, we define a relation: a/b c/d if and only if ad = bc, then is an equivalence relation. 3. If on the integers set, Z, we define a relation: x y if and only if xy 0, then is not an equivalence relation. Show it!
9 I. Set and Function antonius cp 4 4. If Z be partitioned, then the relation is in a same partition class with is an equivalence relation on Z. Definition Let is an equivalence relation on a set A, and x is an element of A. Set of all elements equivalent to x is called equivalence class of x, and be denoted by [x]. It is formally written as [x] = {a A a x}. Theorem If x y then [x] = [y]. Prove the theorem above using the principal of the equality of two sets, i.e it has to be shown that [x] [y] and [y] [x]. Theorem If is an equivalence relation on a set A, then the set of all equivalence classes, {[x] x A}, forms a partition on A. For every n Z + there is an important equivalence relation on Z that is called as congruence modulo n. Definition Let a and b be two integers on Z and n be any positive integer. a is congruent to b modulo n, and be denoted a b (mod n), if a b is evenly divisible by n, so that a b = nk, for some k Z. Equivalence classes for congruence modulo n are residue classes modulo n. Example We see that 7 12 (mod 5) since 7 12 = 5( 1). Residue class containing 7 and 12 is {5n + 2 n Z} = {..., 13, 8, 3, 2, 7, 12, 17,...} 1.3 Function Definition A function φ from a set A to a set B is a relation that assigns all elements of A into exactly one element of B.
10 I. Set and Function antonius cp 5 As notation, φ : A B is a function if ( a A)(!b B), φ(a) = b. Therefore, to show that a relation is a function, it needs to be proved that ( a 1, a 2 A), a 1 = a 2 = φ(a 1 ) = φ(a 2 ) Definition A function from a set A to a set B is injective if for each element of B there is at most one element of A that be connected to it; and is surjective if for each element of B there are at least one element of A that be connected to it. Technically, proving on that two kinds of function can be described below. 1. To show that φ is injective, it has to be proved that φ(a 1 ) = φ(a 2 ) implies a 1 = a To show that φ is surjective, it has to be proved that for every b B, there exist a A such that φ(a) = b. 1.4 Binary Operation Definition Binary operation on a set, S, is a function that assigns each ordered pair of elements of S, (a, b) to an element of S. This definition show that the set S has to be closure under a binary operation. It means that if a, b S and is binary operation on S such that a b = c then c S. Besides that the term ordered pairs has an important role, since element that be connected to (a, b) is not necessary same as the element that be connected to (b, a). Example Addition and multiplication on the set (R), on the set (Z), on the set (C), or on the set (Q) are binary operation. 2. Let M(R) be the set of all matrices with real entries, then usual matrix addition is not a binary operation. Why?
11 I. Set and Function antonius cp 6 3. The usual addition is not a binary operation on the set R = R {0}. Why? For identifying a binary operation on a set S, there are two things should be considered, that is for every ordered pairs (a, b) in S, 1.!c such that a b = c; 2. c S. Definition A binary operation on a set S is commutative if and only if a b = b a, a, b S; and associative if and only if (a b) c = a (b c), a, b, c S. Example Addition and multiplication on the set (R), on the set (Z), on the set (C), or on the set (Q) are commutative and associative. 2. Let M 2,2 (R) is the set of all 2 2 matrices with real entries, then matrix multiplication on M 2,2 (R) is associative but not commutative. Please inspect this! 3. Substraction on R is not associative and also not commutative. Why? 1.5 Exercise on Set and Function 1. If A = {1, 2, 3, 4}, how many subsets of A? Mentione it! 2. Prove that: (a) If M φ, then M = φ. (b) If K L, L M dan M K, then K = M. (c) A (A B) (d) If A B = φ then A = φ dan B = φ. (e) (A B) A (f) A B if and only if (A B) = B (g) (A B) A
12 I. Set and Function antonius cp 7 (h) (A B) B = φ (i) M N if and only if M N = φ (j) M = N if and only if M N = φ dan N M = φ 3. Describe all elements of the following sets. (a) {x R x 2 = 3} (b) {a Z a 2 = 3} (c) {x Z xy = 60 for some y Z} (d) {x Z x 2 x < 115} 4. Determine whether the following relation is an equivalence relation? If yes, describe the partition constructed by the equivalence relation! (a) x y on the set Z if xy > 0 (b) x y on the set R if x y (c) x y on the set R if x = y (d) x y on the set R if x y 3 (e) x y on the set Z + if x and y have the same digits. (f) x y on the set Z + if x and y have the same last digit. (g) x y on the set Z + if n m is divisible by Let n be any integer in Z +. Show that the congruence modulo n is an equivalence relation on Z. 6. Describe all residue classes on Z modulo n, for n = 1,2,3,4 or 8 7. Compute all possible partition on a set S containing 1,2,3,4 or 5 elements. 8. If function f : A B has inverse function f 1 : B A, mention all properties of f. 9. If A = [ 1, 1] and function f 1 (x) = x 2, f 2 (x) = x 3, f 3 (x) = sin x, f 4 (x) = x 5, f 5 (x) = φ x, observe which function that has inverse function!
13 I. Set and Function antonius cp Prove that f : A B and g : B C have inverse function f 1 : B A and g 1 : C B, then function composition g f : A C is also has inverse function f 1 g 1 : C A 11. Let f : A B and g : B A and g f = I A, where I A is the identity function on A. Determine whether the following statement is true or false. (a) f is injective. (b) g is injective. (c) g = f 1 (d) g is surjective. (e) f is surjective. 12. Determine whether the following binary operator is commutative or associative. (a) defined on Z by a b = a b (b) defined on Q by a b = ab + 1 (c) defined on Q by a b = ab 2 (d) defined on Z + by a b = 2 ab (e) defined on Z + by a b = a b 13. Let a set S has exactly one element. How many different binary operation can be defined on S? exactly 3 elements; exactly n elements. Answer the question if S has exactly 2 elements; 14. Determine whether the following operation is binary or not. If not, describe which axiom that is not covered. (a) On the set Z +, define by a b = a b. (b) On the set Z +, define by a b = a b. (c) On the set R, define by a b = a b.
14 I. Set and Function antonius cp 9 (d) On the set Z +, define by a b = c, where c is the smallest integer greater than botha and b. (e) On the set Z +, define by a b = c, where c is at least 5 more than a + b. (f) On the set Z +, define by a b = c, where c is the largest integer less than ab. 15. Determine the following true or false and give the reason. (a) If is a binary operation on a set S, then a a = a, a S. (b) If is a commutative binary operation on a set S, then a, b, c S, a (b c) = (b c) a. (c) If ia an associative binary operation on a set S, then a, b, c S, a (b c) = (b c) a. (d) A binary operation in a set S is commutative if there exists a, b S, such that a b = b a. (e) Every binary operation defined on a set with exactly 1 element is commutative and associative. (f) A binary operation on a set S assigns at least 1 element of S to each ordered pair of elements of S. (g) A binary operation on a set S assigns at most 1 element of S to each ordered pair of elements of S. (h) A binary operation on a set S assigns exactly 1 element of S to each ordered pair of elements of S. 16. Show that if be commutative and associative binary operation on a set S, then (a b) (c d) = [(d c) a] b, a, b, c, d S. 17. Determine the following true or false and give the reason. (a) Every binary operation on a set with one element, is commutative and associative.
15 I. Set and Function antonius cp 10 (b) Every commutative binary operation on a set with exactly 2 elements, is associative. (c) If F is the set of all real function, then function composition on F is commutative. (d) If F is the set of all real function, then function composition on F is associative. (e) If F is the set of all real function,, then function addition on F is associative. (f) If dan be any binary operation on a set S, then a (b c) = (a b) (a c), a, b, c S
16 Chapter 2 GROUP This chapter provides opening material of the first part of the substance of the subject of Algebra Structure (also be known as Abstract Algebra). It contains the definition, properties, order, and concepts of group and its elements. 2.1 Definition of Group An algebra structure is a system that consists of two components, that is a set and a binary operation defined on the set. A system that consists of a non empty set G and a binary operation defined on the set is called grupoid. If the binary operation on the grupoid is associative, then the system is called by semi group. If the semi group consists of identity element, e, such that for every a G satisfy a e = e a = a, then the system is called monoid. And if each element of monoid has own inverse, that is for each a G, a 1 G such that a a 1 = a 1 a = e, then the system is now calledgroup. Now we go the formal definition of group. Definition A group < G, > is a set G, together with a binary operation on G, such that the following axioms are satisfied: 1. G is closed under the operation. That is, a, b G, a b G. 2. The binary operation is associative, that is, ( a, b, c G), (a b) c = a (b c). 11
17 II. Group antonius cp There is an identity element, e, in G. That is, ( e G), ( a G), a e = e a = a. 4. Each element of G has own inverse, that is ( a G), ( a 1 G), a a 1 = a 1 a = e, where e is the identity element in G. Example The set R under usual addition operation, form a group. 2. Z 5 = {0, 1, 2, 3, 4} under addition operation modulo 5, form a group. 3. {a + b 3 a, b Z} under addition defined as follow: (a 1 + b 1 3) + (a2 + b 2 3) = (a1 + a 2 ) + (b 1 + b 2 ) 3, form a group. 4. The set of all 2 2 real matrices can not form a group under matrix multiplication. Why? 5. The set Z can not form a group under multiplication. Why? 6. The set Q under multiplication, form a group. 7. Define the operation, so that G = {a, b, c, d} forms a group. Definition A group < G, > is commutative if ( a, b G), a b = b a. Example The set of integers forms a commutative group under usual addition operation. 2.2 Properties of Group On understanding of the concepts of group, this section provides some basic properties of group. The proof of some theorems will be left as exercise. Theorem The identity element in a group is unique. Proof: If identity element is not unique then there are e and e and both of them are identity elements. If e as the identity element then e e = e. If e as the identity element then e e = e. Since is a binary operation then e = e.
18 II. Group antonius cp 13 Theorem The inverse of each element of a group is unique. Proof: If the inverse of a is not unique, then there are two different elements b and c, where both of them are inverse of a. So that (b a) c = c dan b (a c) = b. Thus, b = c. Theorem If G is a group under binary operation, then G satisfies the left cancelation law and the right cancelation law. That is, a b = a c implies b = c, and a b = c b implies a = c, a, b, c G. Proof for the left cancelation law: a b = a c = a 1 a b = a 1 a c = e b = e c = b = c Using the analog way, prove the right cancelation law! Theorem If G is a group and a 1, a 2,, a n is any n elements of G, then. (a 1 a 2 a n ) 1 = a 1 n a 1 n 1 a 1 1 Theorem If G is a group then for all element a in G (a 1 ) 1 = a. Theorem In a group G, the equation ax = b, where a, b G dan x is a variable, has unique solution, that is x = a 1 b. Theorem If an empty set G under binary operation satisfy the axioms: closed, associatif, and the equation a x = b and y a = b have solution for every a, b G, then (G, ) is a group. 2.3 Order of Group and Element This section provides the definition of order group and elements and its properties.
19 II. Group antonius cp 14 Definition The result of operation of m factors, a a a a a is represented by a m ; The result of operation of m factors, a 1 a 1 a 1 a 1 a 1 is represented by a m ; and a 0 = e, where e is the identity element in G. Theorem If m is a positif integer then a m = (a 1 ) m = (a m ) 1 Example In the group (Z, +), 4 7 = = 28; 4 1 = 4 so that 4 5 = ( 4) + ( 4) + ( 4) + ( 4) + ( 4) = 20; 4 0 = 0, since 0 is the identity element under addition of integers. 2. In the group (R, ), 2 3 = = 8; 2 1 = 1 so that = = 1; = 1, since 1 is the identity element under multiplication of real numbers. Theorem If m and n are integers then a m a n = a m+n and (a m ) n = a mn Definition The order of a finite group G is the number of elements of G. If the number of elements of G is infinite, then the order of G is infinite. The order of G is denoted as G. Definition Let a is an element of a group G. The order of a is n if and only if n is the least positive integer such that a n = e, where e is identity element in group G. If there is no such positive integer, then the order of a is infinite. The order of a is denoted O(a). Example In the group (Z 5, +), O(2) = 5, since 5 is the least positive integer such that 2 5 = (mod5). 2. In the group (Z 5, ), O(2) = 4, since 4 is the least positive integer such that 2 4 = (mod5). Theorem Let a be an element of a group G. If the order of a is n then there exist n variation of power of a in G, they are a 1, a 2, a 3,, a n 1, a n Note that the power here depends on the defined binary operation. Example In (Z 5, +), O(2) = 5, so that there exist 5 variation of power of 2, that is 2 1 = 2; 2 2 = 4; 2 3 = 1; 2 4 = 3; dan 2 5 = 0.
20 II. Group antonius cp In (Z 5, ), O(2) = 4, so that there exist 4 variation of power of 2, that is 2 1 = 2; 2 2 = 4; 2 3 = 3; dan 2 4 = 1. Theorem If the order of a is infinite then all power of a are distinct, that is if r s than a r a s. Example In the (Z, +), O(2) is infinite, so that every power of 2 is different. Prove the following two theorem and give an example for each. Theorem Let O(a) = n. (a k = e) n k (n is a factor of k). Theorem Let O(a) = n then O(a 1 ) = n 2.4 Subgroup Definition Let < G, > be a group and H be a non empty subset of G. H is a subgroup of G if and only if < H, > is also a group. Base on the definition, a subgroup is a group inside the other group. Furthermore, since H is a subset of G then the axiom associative also works on H. Theorem Let < G, > be a group and H be a non empty subset of G. H is a subgroup of G if it satisfy these three axioms. 1. Closed. 2. Identity element 3. Inverse. It also can be analyzed that if the system satisfy axiom closed and inverse, than the identity element will be automatically satisfied. Therefore the theorem above can be simplify to the following theorem.
21 II. Group antonius cp 16 Theorem Let < G, > be a group and H be non empty subset of G. H is a subgroup of G if it satisfy the two axioms below. 1. Closed. That is ( c, d H), c d H. 2. Inverse. That is ( c H), c 1 H. Finally, those two axioms in above theorem can be combined to achieve the following theorem. Theorem Let < G, >be a group and H be a non empty subset of G. H is a subgroup of G if ( c, d H), c d 1 H. Example Both of {0, 3} and {0, 2, 4} are subgroup of Z 6, +). Show it! Definition Let < G, > be a group. Both of H and K are subset of G. Then H K = {a G a = h k, h H k K} and H 1 = {a G a = h 1, h H} Those definition can be used on proving the next theorems. Theorem If < H, > is a subgroup of a group < G, >, then H H = H and H 1 = H. Theorem If both of H and K are subgroup of a group < G, >, then H K is also a subgroup if and only if H K = K H. Theorem If both of H and K are subgroup of a group (G, ), then H K is also a subgroup on < G, >. Theorem Let G be a group and a G. If H is the set of all power of a in G, then H is a subgroup of G. Example In the (Z 5, ), there are 4 variation of power of 2, that is 2 1 = 2; 2 2 = 4; 2 3 = 3; and 2 4 = 1, so that {1, 2, 3, 4} is a subgroup of (Z 5, ).
22 II. Group antonius cp The Exercise for Concepts of Group 1. Determine whether the following algebra structure is a grupoid, semigrup, monoid or group! (a) The set of natural numbers under usual addition. (b) The set of natural numbers under usual multiplication. (c) The set of integers under usual addition. (d) G = {ma a Z} under integers addition, where m is any integer. (e) G = {m a a Z} under multiplication, where m is any integer. (f) G = {a + 2b a, b Q} under addition. (g) The set of non zero rational numbers under multiplication. (h) The set of non zero complex numbers under multiplication. (i) {0, 1, 2, 3,...} under addition. (j) The set M 2 (R) under addition matrix. (k) The set M 2 (R) under multiplication matrix. (l) The set of all integers divisible by 5, under addition. (m) The set of all vectors in R 2 of the form (x, 3x) under addition vector. (n) The set of all vectors in R 2 of the form (0, y) or (x, 0) under addition vector. (o) G = {f 1, f 2, f 3, f 4 } under transformation composition, where f 1 (z) = z, f 2 (z) = z, f 3 (z) = 1, f z 4(z) = 1, for every complex numbers, z. z 2. Let S be the set of all real numbers except 1. On S define by a b = a + b + ab. (a) Show that is a binary operation. (b) Show whether S is a group or not. (c) Compute the solution of 2 x 3 = 7 in S.
23 II. Group antonius cp If G is a group with binary operation, show that a, b G, (a b) 1 = b 1 a 1! 4. Determine true or false the following statements and give the reason. (a) A group may has more than one identity element. (b) On a group, every linear equation has solution. (c) Every finite group containing at most 3 elements is abelian. (d) The empty set can be considered as a group. 5. If G is a finite group with identity element e and of even order, show that there exists a e in G such that a a = e! 6. Let be a binary operation on a set S, an element x S is called idempotent for if x x = x. Show that a group has exactly one idempotent element. 7. If G is a group with identity element eand x G, x x = e, show that G is an abelian group. 8. Show that if (a b) 2 = a 2 b 2, for a and b in G, then a b = b a! 9. Let G be a group and a, b G. Show that (a b) 1 = a 1 b 1 if and only if a*b = b*a! 10. Determine true or false the following statements and give the reasons. (a) Associative law is always satisfied on every group. (b) There may exist a group which is not satisfy the cancelation law. (c) Every group is a subgroup of itself. (d) Every group has exactly two improper subgroup. (e) Every set of numbers that is a group under addition, is also a group under multiplication. (f) Every subset of a group is a subgroup under the same binary operation. (g) If b 2 = e then b = e, where e is the identity element.
24 II. Group antonius cp 19 (h) If c 2 = c then c = e. (i) On every group, a n b n = (a b) n. 11. If both of H and K are subgroup of an abelian group G, show that HK = {hk h H, k K} is also subgroup of G. 12. Show that a non empty subset H of a group G is a subgroup of G if and only if ab 1 H, a, b H! 13. Show that if G is an abelian group with identity element e, then all element x G satisfying the equation x 2 = e forms a subgroup H of G! 14. Let G be a group and a is an element in G. Show that H a = {x G xa = ax} is a subgroup of G! 15. Let H be a subgroup of a group G. For a, b G, let a b if and only if ab 1 H. Show that is an equivalence relation on G. Describe the partition formed by such equivalence relation. 16. Show that if H is a subgroup of G and K is a subgroup of G, then H K is also a subgroup of G!
25 Chapter 3 CYCLIC GROUP As discussed before that the set of all power of an element of a group forms a subgroup. It derives the concept of a group containing all power of an element in it. Such group is then called cyclic group. 3.1 Concept and Basic Properties In the end of previous chapter, it has been described that the set of all power of an element in group forms subgroup. Let G be a group and a G, then H = {h G h = a k, k Z} is a subgroup of G. Therefore a is also in H since a = a 1 and all elements of H can be presented as a power of a. It can be said that a generates the set H, which is a group. This concept is a basic of forming cyclic group. Definition A group G is cyclic if there exists elements a G such that every element x G, can be represented by x = a m, where m is integer. The element a is called by generator and G is a cyclic group developed by a and denoted : G =< a >. Theorem Every cyclic group is abelian. Theorem If G =< a > and b G then O(b) O(a). 20
26 III. Cyclic Group antonius cp 21 Proving on the next theorems of cyclic group need division algorithm on integers (Z). This algorithm is based on the theorem that if b is a positive integer and a be any integer, then there exists unique integers q dan r such that a = bq + r where 0 r < b. Theorem Every subgroup of cyclic group is cyclic. Corollary Subgroups of Z under addition are precisely the groups nz under addition for n Z. Definition Let r and s be two positive integers. The positive generator /(d/) of the cyclic group G = {nr + ms n, m Z} under addition is the greatest common divisor (gcd) of r and s. To understanding that definition, firstly we have to show that G is a subgroup of Z. This is easy to do. Since (Z, +) =< 1 > then G is also cyclic and has a positive generator d. Note from the definition that d is a divisor of r and s since both r = 1r + 0s and s = 0r + 1s are in G. Since d G, then it can be written as d = nr + ms for any integers n and m. It can be shown that every integer that divides r and s will also divides d. Therefore d is the greatest common divisor of r and s. Theorem If G =< a > is of order non prime n, then every proper subgroup of G is generated by a m where m is proper divisor of n. In converse, if m is a proper divisor of n then G has a proper subgroup generated by a m. Theorem If G =< a > and O(a) = n then G = n. Theorem If G =< a > then G =< a 1 >.
27 III. Cyclic Group antonius cp Subgroup of Finite Cyclic Group Let G =< a >, then G may be finite or infinite. 1. If G is infinite, then all power of a are different. Prove it! 2. If G is finite and of order n, then there are exactly n different power of a. Why? Definition Let n be a fixed positive integer and let h and k be any integers. The remainder r when h + k is divided by n in accord with the division algorithm is the sum of h and k modulo n. Example = 31 = 5(6) + 1. So that (mod 5). Theorem The set {0, 1, 2, 3,, n 1} is the cyclic group Z n under addition modulo n. Theorem Let G =< a > and G = n. generate a cyclic subgroup H of G containing n d If b G and b = a s, then b elements, where d is gcd of n and s. Example Consider Z 12 with generator a = 1. Since8 = 8 1 and gcd of 12 and 8 is 4 then 8 generate a subgroup containing 12 4 = 3 elements, that is < 8 >= {0, 4, 8}. Theorem If G =< a > and G = n then the other generators for G are of the forms a r, where r is relatively prime to n. Example as we know that Z 12 =< 1 >. Then the other generators for Z 12 are 5 = 5 1, 7 = 7 1, dan 11 = Exercises on Cyclic Group 1. Prove that every cyclic group is abelian! 2. Show that a group having no proper non trivial subgroup is a cyclic group!
28 III. Cyclic Group antonius cp Compute the gcd of 32 and 24, 48 and 88, 360 and Let + n be notation of addition modulo n. Compute: , , , and Compute the number of generator of cyclic group of order : 5, 8, 12, and Compute the order of a cyclic subgroup of Z 30 generated by Find all subgroup of : Z 12, Z 36, and Z 8, and draw their lattice diagram. 8. Determine all possibilities order of subgroups of the group : Z 6, Z 8, Z 12, Z 60, and Z Determine true or false the following statement and give the reason! (a) In every cyclic group, every element is a generator. (b) Z 4 is a cyclic group. (c) Every abelian group is cyclic. (d) Q under addition is cyclic. (e) Every element of cyclic group is generator. (f) There is at least one abelian group of every finite order > 0. (g) All generators of Z 20 are prime numbers. (h) Every cyclic group of order > 2 has at least two distinct generators. 10. Give an example of a group with the property described, or explain why no example exists. (a) A finite group that is not cyclic. (b) An infinite group that is not cyclic. (c) A cyclic group having only one generator. (d) An infinite cyclic group having four generators. (e) A finite cyclic group having four generators.
29 III. Cyclic Group antonius cp If G =< a > and o(a) = n, prove that G = n. 12. If G =< a >, prove that G =< a 1 >. 13. If a group of order n containing an element of order n, prove that the group is cyclic. 14. In a cyclic group of order n, show that there exists element of order k, where k is a factor of n. 15. Show that a group having finite number of subgroup, is finite. 16. Let p and q be two prime numbers. Compute the number of generators of cyclic group Z pq. 17. Let p be a prime number. Compute the number of generators of cyclic group Z p r, where r is an integer Show that in a finite cyclic group Gof order n, the equation x m = e has exactly m solutions in G for each positive integer m that divides n. 19. Show that Z p has no proper subgroup if p is a prime number. 20. Prove that the order of a cyclic group same as the order of its generator. 21. How many generators of a cyclic group of order 10?
30 Chapter 4 GROUP OF PERMUTATIONS In this chapter we describe a group containing permutations defined in a set. Such group has special characteristics. 4.1 Permutation Definition Permutation in a set A is a one to one function from A onto A. Example Let A = {x, y, z} then α = a b c c a b is a permutation, where α(a) = c; α(b) = a; and α(c) = b. 2. Let B = {1, 2, 3, 4, 5}. Given two permutations in B, φ = and β = then permutation multiplication (= function composition) φβ is : φβ = =
31 IV. Group of Permutations antonius cp 26 Theorem Let A be a non empty set and S A be the set of all permutation in A. Then S A is a group under permutation multiplication. Definition Symmetric Group. Let A = {1, 2, 3,, n}, then group of all permutations on A is called symmetric group n, and be denoted as S n. Note : S n has n! elements. Example Let A = {1, 2, 3} then S 3 has 3! = 6 elements. All permutations on A can be described below. ρ 0 = 1 2 3, µ 1 = 1 2 3, ρ 1 = 1 2 3, µ 2 = 1 2 3, ρ 2 = 1 2 3, µ 3 = 1 2 3, It can be proved that S 3 = {ρ 0, ρ 1, ρ 2, µ 1, µ 2, µ 3 } is a group under permutation multiplication. 4.2 Orbit and Cycle Every permutation σ of a set A determines a natural partition of A into cells with the property that a, b A are in the same cell if and only if b = σ n (a), for some n Z. It can be proved that relation defined by a b b = σ n (a), is a equivalence relation. Definition Let σ be a permutation of a set A. The equivalence classes determined by the equivalence relation a b b = σ n (a) are the orbits of σ.
32 IV. Group of Permutations antonius cp 27 Example The orbits of permutation σ = of S 8 can be found by applying σ repeatedly, obtaining symbolically hence the orbits of σ are {1, 2, 8}, {3, 4, 6}, {5, 7} Each orbit in the example above determines a new permutation in S 8 by acts on the orbit members and leaves the remaining elements fixed. For example the orbit {1, 2, 8} with arrow direction forms the permutation µ = The permutation µ has only 1 orbit containing more than 1 element. Such permutation is called cycle. Let say the formal definition. Definition A permutation σ S n is a cycle if σ has at most one orbit containing more than one element. The length of a cycle is the number of elements in its largest orbit. Example As mentioned in previous example, permutation µ is a cycle of length 3 and be denoted as µ = (1, 8, 2)
33 IV. Group of Permutations antonius cp 28 Note, unlike in the orbit notation, the order of elements in the cycle notation determines moving flow. For example, (1, 8, 2) = (8, 2, 1) = (2, 1, 8) but (1, 8, 2) (1, 2, 8). As described before that the set of orbits of a permutation is a partition on S n, therefore the orbits of a permutation are disjoint sets. Furthermore, since an orbit determines a cycle, then we can derive the following theorem. Theorem Each permutation σ in a finite set is a product of disjoint cycles. Example σ = = (1, 8, 2)(3, 6, 4)(5, 7) Definition A cycle of length 2 is called transposition. Each cycle can be described as a product of transpositions, (a 1, a 2, a 3,, a n 1, a n ) = (a 1, a n )(a 1, a n 1 ) (a 1, a 3 )(a 1, a 2 ). Therefore, a permutation is also a product of transpositions. Theorem Let σ S n and τ be a transposition on S n. orbits of σ and the number of orbits of τσ differ by 1. The number of Let τ = (i, j), then the above theorem can be proved by analyzing the possibilities below. 1. i and j are in the different orbits of σ; 2. i and j are in the same orbit of σ. Definition 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 transposition or the product of an odd number of transposition, respectively. Example Determine whether even or odd the following permutations.
34 IV. Group of Permutations antonius cp σ = 2. µ = Definition The subgroup of S n consisting even permutations is called alternating group, A n on n letters. Example Determine the alternating group of S 3! 4.3 Exercises on Group of Permutation 1. Compute the number of elements of the set below. (a) {α S 4 α(3) = 3}. (b) {α S 5 α(2) = 5}. 2. Determine true or false the following statements and give the reason. (a) Every permutation is a one to one function. (b) Every function is permutation if and only if it is one to one. (c) Every function from a set onto itself should be one to one. (d) The symmetric group S 10 has 10 elements. (e) The symmetric groups 3 is cyclic. (f) S n is not cyclic for each n. 3. Show that the symmetric group S n is not abelian for n Find all orbits of the following permutations: (a) α : Z Z, where α(n) = n + 1. (b) α : Z Z, where α(n) = n + 2. (c) α : Z Z, where α(n) = n 3.
35 IV. Group of Permutations antonius cp For permutation on {1, 2, 3, 4, 5, 6, 7, 8}, compute the product of cycles below. (a) (1, 4, 5)(7, 8)(2, 5, 7) (b) (1, 3, 2, 7)(4, 8, 6) (c) (1, 2)(4, 7, 8)(2, 1)(7, 2, 8, 1, 5) 6. Express the following permutations as a product of disjoint cycles, and then express them as product of transpositions. (a) (b) (c) Determine true or false the following statements and give the reason. (a) Every permutation is a cycle. (b) Every cycle is a permutation. (c) Alternating group A 5 has 120 elements. (d) Alternating group A 3 is a commutative group. (e) The symmetric group S n is not cyclic for every n Show that if H is a subgroup of S n, for n 2, then all permutation in H is even or exactly half of them are even. 9. Let G be a group and a be a fixed element in G. Show that map λ a : G G, where λ a (g) = ag, g G, is a permutation in G.
36 Chapter 5 COSET AND THE LAGRANGE THEOREM A subgroup of a group has cosets. The set of the cosets determines a partition on the group. Furthermore, there is a one to one correspondence between the subgroup and its coset. This concept leads the relationship between the order of a group and the order of subgroup. 5.1 Coset Theorem Let H be a subgroup of a group G. The relation L determined by a L b a 1 b H and the relation R determined by a R b ab 1 H are an equivalence relation. Prove the theorem above! The relation L or R determine equivalence classes in G. For example, the class containing a formed by L is [a] = {x G a L x} or equivalently equal to {x G a 1 x H}, or equivalently equal to {x G a 1 x = h, h H}, or equivalently equal to {x G x = ah, h H}. So that [a] = {ah h H} = ah. Please observe the class containing a formed by R. 31
37 V. Coset and the Lagrange Theorem antonius cp 32 Definition Let G be a group and H be a subgroup of G. For any element a G; ah = {x G x = ah, h H} is called left coset of H containing a; and Ha = {y G y = ha, h H} is called right coset of H containing a. Example Z is a group under integer addition and 3Z= {, 6, 3, 0, 3, 6, 9, } is a subgroup of Z. The left cosets of 3Z in Z are 3Z= {, 6, 3, 0, 3, 6, 9, } 1 + 3Z= {, 5, 2, 1, 4, 7, 10, } 2 + 3Z= {, 4, 1, 2, 5, 8, 11, } It can be shown that {3Z, 1 + 3Z, 2 + 3Z} is a partition of Z. Whether we use left coset or right coset, there is no significantly different,as long as we are consistent. In the next description we use left coset, which is analog to the right coset. Here are some theorems regarding the coset. Theorem Coset ah = bh if and only if a bh. Theorem Let G be a group and H be a subgroup of G. The family set of all coset of H is a partition of G. Theorem If ah is a coset of H, then H and ah are one to one correspondence. 5.2 The Lagrange Theorem If G be a group and H be a subgroup of G, then the set of all cosets of H forms a partition on G and H is one to one corresponds to each coset. For finite set, the concept of one to one correspondence indicates the equality of the number of elements. Therefore, in a finite group, there is a relationship between the group order and its subgroup order. Theorem The Lagrange Theorem. If G be a finite group and H subgrup of G, then the order of H is a factor of the order of G.
38 V. Coset and the Lagrange Theorem antonius cp 33 Example If G is of order 8, then the possible order for H are 1, 2, 4, or 8. If H = 1 then H = {e}, where e is the identity element of G. If H = 8 then H = G. 2. It can be observe that subgroups of Z 6 = {0, 1, 2, 3, 4, 5} are {0}, {0, 3}, {0, 2, 4}, dan Z 6 itself. The following are some theorems as corollary of the Lagrange theorem. Theorem If G be a prime ordered group then G is a cyclic group and each element except the identity element is a generator. Theorem The order of each element of a finite group is a factor of the order of the group. Definition If G be a group and H be a subgrup of G then index of H is the number of cosets of H in G, and be denoted. Therefore, a finite group satisfies (G : H) (G : H) = G H 5.3 Exercises on Coset and the Lagrange Theorem 1. Find all cosets of the following subgroups. (a) 4Z of group Z. (b) 4Z of group 2Z. (c) < 2 > of group Z 12. (d) < 4 > of group Z 12.
39 V. Coset and the Lagrange Theorem antonius cp 34 (e) < 18 > of group Z 36. (f) {ρ 0, ρ 2 } of group D 4. (g) {ρ 0, µ 2 } of group D Compute the index of each subgroup below. (a) < 3 > of group Z 24. (b) < µ 1 > of group S 3. (c) < µ 3 > of group D Determine true or false the following statements and give the reason. (a) Every subgroup of every group has coset. (b) The number of coset of subgroup in a finite group divides the order of the group. (c) Every group of prime order is abelian. (d) A subgroup is a coset for itself. (e) There only subgroup of a finite group that has coset. 4. Let H be a subgroup of a group G such that g 1 hg H, g G and h H. Show that each left coset of H, gh, same as the right coset, Hg! 5. Let H be a subgroup of a group G. Show that if the partition on G by the left coset of H same as the partition on G by the right coset of H, then g 1 hg H is satisfied, g G and h H! 6. Give the proof if the following is true, or give acounterexample if it is false. (a) If ah = bh then Ha = Hb (b) If Ha = Hb then b Ha (c) If ah = bh then Ha 1 = Hb 1 (d) If ah = bh then a 2 H = b 2 H
40 V. Coset and the Lagrange Theorem antonius cp If G be a group of order pq, where both of p and q are prime, show that each proper subgroup of G is cyclic. 8. Show that a group having at least two elements but no proper non trivial subgroup, must be a finite group of prime order. 9. Show that if H be a subgroup of index 2 in a finite group G, then each left coset of H is also its right coset. 10. Show that if a group G with identity elemente and of finite order, n, then a n = e, a G! 11. Let H and K be two subgroup of a group G. Define a relation in G determined by a b if and only if a = hbk, for any h H dan k K. (a) Prove that is an equivalence relation in G! (b) Describe all elements of the equivalence class containing a!
41 Chapter 6 GROUP HOMOMORPHISM This chapter is the beginning of second part of the subject of Algebra Structure. In the first part, we discuss about group and its properties. In this second part we will discuss about a kind of function that assigns a group to another group. This chapter provides the definition of homomorphism and its properties. 6.1 Homomorphism Definition A map φ from a group (G, ) into a group (G, #) is called homomorphism if φ(a b) = φ(a)#φ(b) for all a and b in G. The equation in the above definition shows a relationship between two binary operations and #, and thus a relationship between two groups G and G. Notes : For simplify the notation we will not write the binary operation symbol. Let G be a group and a, b G then binary operation on a and b be written as ab. Between any two groups G and G, there is at least a homomorphism φ : G G called trivial homomorphism defined by φ(g) = e for each g G where e is the identity element of G. Example Let α : Z Z n defined by α(m) = r, where r is the remainder when m is divided by n. Then α is a homomorphism. 36
42 VI. Group Homomorphism antonius cp 37 Example Let S n be a symmetric group on n letters, and let φ : S n Z 2 defined by 0 if ρ is an even permutation, φ(ρ) = 1 if ρ is an odd permutation Then φ is a homomorphism. Prove it!! Example Let F be an addition group of all function in R, and R be an addition group of real numbers, and let c be a real number. Let φ c : F R defined by φ c (f) = f(c) for each f F. Then φ c is a homomorphism and is called evaluation homomorphism. Definition Let φ be a mapping of a set X into a set Y, and let A X and B Y. The image of A in Y is φ(a) = {φ(a) a A}. The set φ(x) is called the range of φ. The inverse image of B in X is φ 1 (B) = {x X φ(x) B} Theorem Let φ be a homomorphism of a group g into a group G, 1. If e is the identity in G, then φ(e) is the identity e in G ; 2. If a G, then φ(a 1 ) = φ(a) 1 ; 3. If H is a subgroup of G, then φ(h) is a subgroup of G ; 4. If S is a subgroup of G, then φ 1 (S ) is a subgroup of G; Loosely speaking, φ preserves the identity, inverses, and subgroups. Definition Let φ : G G be a homomorphism, then the kernel of φ, denoted by Ker(φ), is defined as Ker(φ) = φ 1 ({e }) = {a G φ(a) = e } where e is the identity of G.
43 VI. Group Homomorphism antonius cp 38 Theorem If φ : G G is a homomorphism, then Ker(φ) is a subgroup of G. Prove it! Theorem Let φ : G G be a homomorphism and H = Ker(φ). Let a G. Then φ 1 {φ(a)} = {x G φ(x) = φ(a)} is the left coset ah of H and is also the right coset Ha of H. Consequently, the two partitions of G into left cosets and into right cosets of H are the same. Corollary A group homomorphism φ : G G is a onetoone map if and only if Ker(φ) = {e}. Prove it! Definition Onetoone homomorphism is called monomorphism, onto homomorphism is called epimorphism. Definition A subgroup H of a group G) is normal if gh = Hg for all g G. 6.2 Isomorphism dan Cayley Theorem Definition An isomorphism φ : G G is a onetoone homomorphism of G onto G. If so then we say that G and G are isomorph and be denoted as G G. Theorem Let ζ be a set of groups. The relation in ζ is a equivalence relation. Prove it! The steps to show two groups that are isomorph is as follows. 1. Define a function φ.
44 VI. Group Homomorphism antonius cp Show that φ is onetoone function. 3. Show that φ is onto. 4. Show that φ(xy) = φ(x)φ(y), x, y G. Theorem All infinite cyclic group, G, are isomorph to the group of integers, Z, under addition. Prove the above theorem by define a function φ : G Z where φ(a n ) = n, a n G. Now, how to show that two groups are not isomorph? To do that we need to know some structural properties, that is the properties that must be shared by any isomorphic groups. Some examples on the structural properties are cyclic; abelian; finite and infinite; group order; the number of elements with certain order; the solution of an equation on a group. The non structural properties, in other hand, may not be shared by any isomorphic groups. Some example of them are as follows. group containing 5; all elements of group are number; the binary operation on a group is a function composition; the elements of a group are permutations; the group is a subgroup of < R, >
45 VI. Group Homomorphism antonius cp 40 Example It cannot be said that Z and 3Z are not isomorph since 17 Z but 17 3Z. This properties is non structural properties. In the fact Z and 3Z are isomorph since the function φ : Z Z, where φ(n) = 3n, is a isomorphism. 2. It can not be said that Z and Q are not isomorph since 1 Q but 1 Z. 2 2 But it can be said that Z and Q are not isomorph since Z is cyclic, but Q is not cyclic. Theorem Cayley Theorem. Every group isomorphs to a group of permutation. Prove the Cayley theorem above by following steps. 1. Beginning with a group G, collect all permutation in G into a set G 2. Prove that G is a group under permutation multiplication. 3. Define function φ : G G and show that φ is an isomorphism of G to the G. 6.3 Exercises on group Homomorphism 1. Determine whether the following mapping is a homomorphism or not and give the reasons. (a) φ : Z R defined by φ(x) = x. (b) φ : R Z defined by φ(x) = x. (c) φ : Z 6 Z 2 defined by φ(x) = the remainder when x is divided by 2. (d) φ : Z 9 Z 2 defined by φ(x) = the remainder when x is divided by 2. (e) φ : Z Z defined by φ(x) = x (f) φ : F F defined by φ(f) = 3f, where F is the group of all real functions. (g) φ : R Z under addition, and be defined by φ(x) = x, x R.
46 VI. Group Homomorphism antonius cp 41 (h) φ : [R, +] [R, ] defined by φ(x) = 2 x (i) φ i : G i G 1 G 2... G i... G r defined by φ i (g i ) = (e 1, e 2,..., g i,..., e r ), dimana g i G i dan e j elemen identitas dalam G j. (j) φ : G G defined by φ(x) = x 1, x G. (k) φ : F F defined by φ(f) = f, f F, where F is the addition group of all functions R R that are differentiable on all order. (l) φ : F R defined by φ(f) = 4 f(x)dx, where F is the addition group 0 of all continue function R R ; R is the addition group of real numbers. (m) φ : F F defined by φ(f) = 3f, where F is the addition group of all function R R. (n) φ : M n R defined by φ(a) = det(a), where M n the addition group of all matrices 2x2, and R is the addition group of all real numbers. 2. Let F = R R be an addition group, D = {f R R f differentiable }. Is φ : f D f F homomorphism or not? If yes, find Ker(φ). 3. Let G be a group. If φ : Z Z G is a homomorphism and let φ(1, 0) = h and φ(0, 1) = k, compute φ(m, n). 4. Let G be a group and g G. Let φ g : x G gx G. Determine g, in orderφ g be homomorphism. 5. Let G be a group and g G. Let φ g : x G gxg 1 G. Determine g, in order φ g be homomorphism. 6. Let φ : G G be a homomorphism. Show that if G < then φ(g) < and φ(g) G. 7. Let φ : G G be a homomorphism. Show that if G < then φ(g) < and φ(g) G. 8. Let φ : G G be a homomorphism. Show that if G be a prime, then φ must be a trivial homomorphism or monomorphism.
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