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Chapter 2 Groups: Introduction, Main De…nitions and Examples We begin with some general statements and practical examples, before looking at precise de…nitions. 2.1 2.1.1 Introduction to Groups Introduction The concept of a group really started in the 1830’s; the term was coined by Evariste Galois in 1830. Today, groups and other algebraic structures are present in every branch of mathematics as well as other …elds such as chemistry, physics to name a few. A group is an algebraic structure consisting of a set and a binary operation. To be a group, the binary operation must satisfy certain properties we will study in the next section. The reader has already encountered groups and used their properties. For example, the set of real numbers with addition is a group. Many examples include sets of numbers of other mathematical objects. There are also more applied examples, we present a few in this section. For example, the possible manipulations of a Rubik’s cube form a group. The examples we will discuss are related to the symmetries of an object, that is the transformations which leave the object unchanged. 2.1.2 An Example: Symmetries of a Square Suppose that we remove a square region from a plane, move it in some way, then put it back in the place it originally occupied. We wish to describe all the ways this can be done. We claim that any motion is equivalent to one of eight possible ways. The square can be placed in four di¤erent positions on two 37 38CHAPTER 2. GROUPS: INTRODUCTION, MAIN DEFINITIONS AND EXAMPLES di¤erent sides. In fact, the eight motions which will accomplish this are listed in the table below: Motion R0 R90 R180 R270 H V D D0 Description Rotation of 0 Rotation of 90 Rotation of 180 Rotation of 270 Flip about a horizontal axis Flip about a vertical axis Flip about the main diagonal Flip about the other diagonal We denote D4 the set of these motions. In other words D4 = (R0 ; R90 ; R180 ; R270 ; H:V; D; D0 ). It is called the dihedral group of order 8. What is even more amazing is that no matter how we combine these motions, we always end up with one of them. The table below, known as a Cayley table after the British mathematician Arthur Cayley, summarizes these results. It works as follows. We combine elements on the left …rst column with elements of the top row. The result is placed where the corresponding row and column intersect. For the time being, let’s denote AB the combination of A and B with the understanding that B is done …rst. This is similar to composition of functions. Indeed, these motions can be thought of as functions from the square region to itself. For example, looking at the table, we see that HR270 = D0 but R270 H = D. R0 R90 R180 R270 H V D D0 R0 R0 R90 R180 R270 H V D D0 R90 R90 R180 R270 R0 D D0 V H R180 R180 R270 R0 R90 V H D0 D R270 R270 R0 R90 R180 D0 D H V H H D0 V D R0 R180 R270 R90 V V D H D0 R180 R0 R90 R270 D D H D0 V R90 R270 R0 R180 D0 D0 V D H R270 R90 R180 R0 We make several remarks about the table: 1. By combining these motions, we only obtain one of the eight existing motions. In other words, we never obtain a new motion. This is one of the conditions for an operation on a set to form a group. This property is called closure. 2. We note that if A 2 D4 , then AR0 = R0 A. In other words, R0 does not change anything. It behaves like 0 for addition of integers or real numbers. Such an element is called an identity element. 3. We see that for each element A of D4 , there is a unique element B satisfying AB = BA = R0 (the identity element). When this happens, B is 2.1. INTRODUCTION TO GROUPS 39 called the inverse of A. For example, the inverse of H is H. The inverse of R90 is R270 . 4. We note that the order in which we combine these motions is important. For example, V R90 = D0 but R90 V = D. So, in general, AB 6= BA if A and B are elements of D4 . When AB = BA for every A and B, we say the group is commutative or Abelian (named after Niels Abel). 5. We see that each element of D4 appears only once in each row or column. We will see it is indeed the case for a group. 6. We see that the table seems pretty orderly. We will see it is indeed the case for a group. The properties outlined above will be de…ned precisely in the next section. They are what makes a group. We will also see that many other properties can be proven from these. 2.1.3 Problems Do # 1, 2, 3 on page 35 of your book. 40CHAPTER 2. GROUPS: INTRODUCTION, MAIN DEFINITIONS AND EXAMPLES 2.2 Main De…nitions, Results and Examples A group, like other mathematical structures you have and/or will study, consists of a set and one or more operations. For a group, there is one operation, called a binary operation because it acts on two elements. If G is a set and an operation, we denote (G; ) the algebraic structure consisting of the set G and the operation . We say that G is a group under or (G; ) is a group. We begin by giving some important de…nitions. 2.2.1 De…nitions De…nition 122 (Binary Operation) A binary operation on a set G is a function which assigns to each ordered pair of elements of G a third element. G is said to be closed under the operation if the third element is also an element of G. Example 123 R is closed under addition. Example 124 Z is closed under addition. Example 125 Z is not closed under division. Example 126 R+ = fx 2 R j x > 0g is closed under multiplication. Example 127 R = fx 2 R j x < 0g is not closed under multiplication. There are several basic properties a binary operations can have. These properties determine what kind of algebraic structure a set together with the binary operation will be. We list them below. De…nition 128 (Properties of Binary Operations) Let G be a set and be a binary operation on G. Let a; b; c be arbitrary elements of G. 1. Commutativity: a b = b a 2. Associativity: a (b c) = (a b) c 3. Right identity: e is a right identity if a e = a 4. Left identity: e is a left identity if e a = a 5. Identity: e is an identity element if a e = e a = a that is a right and a left identity. 6. Right inverse: a has a right inverse we will call a0 if a a0 = e where e is an identity element. 7. Left inverse: a has a left inverse we will call a0 if a0 an identity element. a = e where e is 2.2. MAIN DEFINITIONS, RESULTS AND EXAMPLES 41 8. Inverse: a has an inverse we will call a0 if a0 is a right and a left inverse of a that is if a a0 = a0 a = e where e is an identity element. Remark 129 The notion of identity and inverse are dependent on the operation. If there is a risk of confusion, we will specify the operation. For example, we may talk about the additive identity or the multiplicative identity. Example 130 The identity element for addition in Z or R is 0. Example 131 If x is any real number, then dition. De…nition 132 (Group) Let G be a set and x is the inverse of x under ad- an operation on G. 1. (G; ) is a group if the following properties are satis…ed: (a) G is closed under (b) is associative (c) has an identity element (d) Every element of G has an inverse under 2. If in addition is commutative, (G; ) is said to be a commutative group or an Abelian group. It is important to note that it only requires four properties for an algebraic structure to be a group. As we will see, many more properties can be derived from these four properties. Hence, every group will satisfy these properties. 2.2.2 Elementary Examples We now look at some examples. It is important to have many examples in mind to illustrate the various properties we will study and also to test conjectures we may have. Example 133 (Z; +), (Q; +) and (R; +) are groups. In each case the identity element is 0. The inverse of a is a. Remark 134 In the above example, we used the terms "the inverse" and "the identity", thus implying these seem to be unique. Indeed they are, we will prove this result below. Example 135 (Z; ) is not a group. Associativity fails. Example 136 Z is not a group under multiplication. The identity would be 1. The inverse property fails. For example, 3 does not have an inverse under multiplication in Z. 42CHAPTER 2. GROUPS: INTRODUCTION, MAIN DEFINITIONS AND EXAMPLES Example 137 R = fx 2 R j x 6= 0g is a group under ordinary multiplication. 1 The identity is 1 and the inverse of a is = a 1 . a Example 138 R+ is also a group under ordinary multiplication. The identity 1 is 1 and the inverse of a is = a 1 . a Example 139 Q and Q+ are also groups under ordinary multiplication. The 1 identity is 1 and the inverse of a is = a 1 . a 2.2.3 Results The properties of a group require the existence of an identity element and also an inverse for each element in the group. Could there be more than one identity? Could an element have more than one inverse? We answer these questions here, and more. Also, let us make a remark about notation. The properties we are about to prove are true for every group, no matter which operation the group has. We will adopt the notation of the multiplication operation. It is important to keep in mind that what we write applies to whatever operation the group has. When we write ab we do not mean a "times" b. Rather, we mean a combined with b using the operation of the group. If the operation is addition, then ab would mean a + b. Theorem 140 (Uniqueness of Identity) In a group G, the identity element is unique. Proof. One way to prove uniqueness is to assume there are two such elements and show they are the same. In light of this, let us assume there are two identity elements e and e0 . We want to show that e = e0 . We know that ae = ea = a and ae0 = e0 a = a for every a 2 G. So, this works in particular for e and e0 . We can write e0 = e0 e since e is an identity element = e since e0 is an identity element Therefore, we can now speak of "the" identity element. Theorem 141 (Cancellation) In a group, the right and left cancellation laws hold, that is: ac = bc =) a = b (right cancellation) ca = cb =) a = b (left cancellation) 2.2. MAIN DEFINITIONS, RESULTS AND EXAMPLES 43 Proof. We only prove the right cancellation law. Suppose that ac = bc and let c0 be the inverse of c. Then, we have ac = bc =) (ac) c0 = (bc) c0 =) a (cc0 ) = b (cc0 ) by associativity =) ae = be =) a=b One consequence of the cancellation law is that in the Cayley table of a group, each group element occurs exactly once in each row and each column. Theorem 142 (Uniqueness of Inverses) Let G be a group. For each a 2 G, there exists a unique inverse that is a unique element b such that ab = ba = e. Proof. Suppose a has two inverses, say b and c. We show we must have b = c. If b and c are inverses of a, then ab = e and ac = e. Therefore ab = ac. By the cancellation law, we must have b = c. In light of these properties, we change our notation as follows: First, recall we will use the multiplicative notation, no matter what the operation is. So, the inverse of a will be denoted a 1 . Again, keep in mind that if the operation is addition, then a 1 is really a. Also, aa:::a will be denoted an (if we have n factors). Again, keep in mind if the operation is addition, aa:::a really means a + a + ::: + a = na. The identity element will usually be denoted e. So, a0 = e. In an abstract group, we do not allow exponents which are not integers. jnj If n is negative, we de…ne an = a 1 The usual laws of exponents apply. am an = am+n n Be careful, if the group is not commutative, (ab) 6= an bn As already mentioned, when dealing with a speci…c operation, we must adapt the notation to it. The table below gives the parallel between multiplication and addition. Multiplicative group a b or ab multiplication e or 1 identity a 1 inverse an ab 1 Additive group a+b addition 0 identity a inverse na a b 44CHAPTER 2. GROUPS: INTRODUCTION, MAIN DEFINITIONS AND EXAMPLES Theorem 143 Let G be a group and a and b elements in G. Then, (ab) 1 =b 1 1 a 1 Proof. Since inverses are unique, and (ab) denotes the inverse of ab, it is enough to verify that b 1 a 1 is the inverse of ab. 1 (ab) b a 1 1 = a bb a 1 1 = aea 1 = aa = e and b 1 a 1 (ab) = b 1 = b 1 eb = b 1 b 1 a a b = e These properties are used when solving equations in a group. Probably until now, all the equations you have solved were with real numbers. You used techniques without even thinking why and how they worked. It is time to change this. Whatever you do when solving an equation, you must understand why you are doing it, if it is even valid. Let us look at some examples. Example 144 Let a, b, c and x be elements of a group G. Solve the equation abxc = b for x. abxc = () () b () abx = bc 1 bx = a x=b 1 bc 1 1 bc a 1 1 There are other important results, which will be proven in the exercises. We now look at more challenging examples. 2.2.4 More Examples Example 145 A rectangular array of the form a c b d is called a 2 2 ma- a b a0 b0 a + a0 b + b0 + = . The 0 0 c d c d c + c0 d + d0 set of 2 2 matrices with real entries is a group under addition. The identity 0 0 a b a b element is and the inverse of is . 0 0 c d c d trix. Addition is de…ned by 2.2. MAIN DEFINITIONS, RESULTS AND EXAMPLES Example 146 Multiplication of matrices is de…ned by a c 45 b d a0 c0 b0 d0 = aa0 + bc0 bd0 + ab0 . he set of 2 2 matrices with real entries is not a ca0 + dc0 dd0 + cb0 group under multiplication. To make it a group, we need to add a restriction. a b We de…ne the determinant of the 2 2 matrix A = , denoted det (A), c d to be det (A) = ad bc. One of the properties of the determinant is that if A and B are two 2 2 matrices then det (AB) = det (A) det (B). Let us de…ne the set GL (2; R), called the general linear group of 2 2 matrices over R by GL (2; R) a c b d j a; b; c; d 2 R and ad bc 6= 0 Then, GL (2; R) is a non-Abelian group under matrix multiplication. The iden3 2 b d 1 0 a b 7 6 tity element is and the inverse of is 4 ad c bc ad a bc 5 0 1 c d ad bc ad bc Example 147 Let Zn = f0; 1; 2; :::; n 1g for n 1. Zn is a group under addition modulo n. Recall that if a 2 Zn and b 2 Zn then (a + b) mod n is the remainder of the division by n in other words if a + b = nq + r then (a + b) mod n = r. Clearly, we have closure since if a 2 Zn and b 2 Zn then (a + b) mod n is the remainder of the division by n and is therefore an integer in Zn . The identity element is 0 and the inverse of j for any j > 0 is n j. This group is called the group of integers modulo n. Example 148 Let U (n) be the set of all positive integers less than n which are relatively prime to n. Then U (n) is a group under multiplication modulo n. Recall that (ab) mod n is the remainder of the division by n in other words if ab = nq + r then (ab) mod n = r. Note that if n is prime, then U (n) = f1; 2; 3; :::; n 1g. For n = 10, U (10) = f1; 3; 7; 9g. We give Cayley’s table for illustration. mod 10 1 3 7 9 1 1 3 7 9 3 3 9 1 7 7 7 1 9 3 9 9 7 3 1 The proof that it is a group is left as an exercise. Example 149 The set Rn = f(a1 ; a2 ; :::; an ) j ai 2 R for i = 1; 2; :::; ng is a group under componentwise addition that is (a1 ; a2 ; :::; an ) + (b1 ; b2 ; :::; bn ) = (a1 + b1 ; a2 + b2 ; :::; an + bn ). Example 150 If F is any of Q, R, C or Zp (p prime), we de…ne the special linear group of 2 2 matrices over F , denoted SL (2; F ), by SL (2; F ) a c b d j a; b; c; d 2 R and ad bc = 1 46CHAPTER 2. GROUPS: INTRODUCTION, MAIN DEFINITIONS AND EXAMPLES It is a non-Abelian group under matrix multiplication. Note that in the case of Zp , we use modulo p arithmetic. 2.3 Problems 1. Prove that if a and n are positive integers then the equation ax mod n = 1 has a solution if and only if 1 = gcd (a; n). 2. Using the above result, prove that U (n) is a group under multiplication modulo n. 3. Write the Cayley table for the operation of each group below: (a) Z5 with addition mod 5. (b) U (5) with multiplication mod 5. (c) U (10) with multiplication mod 10 a 4. Let G = fa; b; cg and be an operation on G de…ned by the table b c Explain why (G; ) is not a group. a a b c b b a b 5. Let a, b, and c be elements of a group. Solve the equation abxc = b for x. 6. Do # 1, 2, 3, 4, 7, 9, 11, 12, 13, 15, 17, 18, 19, 23, 25, 26, 27, 35, 36 on pages 52-54. c c . c a