(Notice also that this set is CLOSED, but does not have an IDENTITY and therefore also does not have the INVERSE PROPERTY.)

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1 Definition 3.1 Group Suppose the binary operation p is defined for elements of the set G. Then G is a group with respect to p provided the following four conditions hold: 1. G is closed under p. That is, x [ G and y [ G imply that x * y is in G. 2. p is associative. For all x, y, z in G, x * (y * z) 5 (x * y) * z. 3. G has an identity element e. There is an e in G such that x * e 5 e * x 5 x for all x [ G. 4. G contains inverses. For each a [ G, there exists b [ G such that a * b 5 b * a 5 e. Definition 3.2 Abelian Group Let G be a group with respect to p. Then G is called a commutative group, or an abelian group, if p is commutative. That is, x * y 5 y * x for all x, y in G. Remark 1) The set of integers is a group under the OPERATION of addition: We have already seen that the integers under the OPERATION of addition are CLOSED, ASSOCIATIVE, have IDENTITY 0, and that any integer x has the INVERSE x. Because the set of integers under addition satisfies all four group PROPERTIES, it is a group! 2) The set {0,1,2} under addition is not a group, because it does not satisfy all of the group PROPERTIES: it does not have the CLOSURE PROPERTY (see the previous lectures to see why). Therefore, the set {0,1,2} under addition is not a group! (Notice also that this set is ASSOCIATIVE, and has an IDENTITY which is 0, but does not have the INVERSE PROPERTY because 1 and 2 are not in the set!) 3) The set of integers under subtraction is not a group, because it does not satisfy all of the group PROPERTIES: it does not have the ASSOCIATIVE PROPERTY (see the previous lectures to see why). Therefore, the set of integers under subtraction is not a group! (Notice also that this set is CLOSED, but does not have an IDENTITY and therefore also does not have the INVERSE PROPERTY.) 4) The set of natural numbers under addition is not a group, because it does not satisfy all of the group PROPERTIES: it does not have the IDENTITY PROPERTY (see the previous lectures to see why). Therefore, the set of natural numbers under addition is not a group! (Notice also that this set is CLOSED, ASSOCIATIVE, but does not have the INVERSE PROPERTY because none of the negative numbers are in the set.) 1

2 5) The set of rational numbers with the element 0 removed is a group under the OPERATION of multiplication: We have already seen that the set rational numbers with the element 0 removed under the OPERATION of multiplication is CLOSED, ASSOCIATIVE, have IDENTITY 1, and that any integer x has the INVERSE 1 x. Because the set of rational numbers with the element 0 removed under multiplication satisfies all four group PROPERTIES, it is a group! 6) The set of rational numbers (which contains 0) under multiplication is not a group, because it does not satisfy all of the group PROPERTIES: it does not have the INVERSE PROPERTY (see the previous lectures to see why). Therefore, the set rational numbers under multiplication is not a group! (Notice also that this set is CLOSED, ASSOCIATIVE, and has an IDENTITY which is 1.) 7) The set of rational numbers under division is not a group, because it does not satisfy all of the group PROPERTIES: it does not have the ASSOCIATIVE PROPERTY (see the previous lectures to see why). Therefore, the set of rational numbers under division is not a group! (Notice also that this set is not CLOSED because anything divided by 0 is not in the set, does not have an IDENTITY and therefore also does not have the INVERSE PROPERTY.) 8) The set of natural numbers under division is not a group, because it does not satisfy all of the group PROPERTIES: it does not have the IDENTITY PROPERTY (see the previous lectures to see why). Therefore, the set of natural numbers under division is not a group! (Notice that this set does not have the CLOSURE, ASSOCIATIVE or INVERSE PROPERTIES.) 9) The set of integers under multiplication is not a group, because it does not satisfy all of the group PROPERTIES: it does not have the INVERSE PROPERTY (see the previous lectures to see why). Therefore, the set of integers under multiplication is not a group! (Notice also that this set is CLOSED, ASSOCIATIVE, and has the IDENTITY ELEMENT 1.) Definition A permutation of a set is a bijection (one to one and onto) of this set onto itself, i.e., S( A) = { f f : A A and f isbijective }. Example* 2

3 Homework*. Take A = { 1,2,3} and obtain an explicit example of S( A). Note that (i) if α : X X is a permutation, than the inverse function α 1 : X X is also a permutation; (ii) if α : X X and β : X X are permutations, one can form a composition α β : X X, which is also a permutation. Theorem Let A be a nonempty set, and let G be collection of all permutations of A. Then G is a group under the composition. Example**. Construct the Cayley (multiplication) table for S(A) in Example*. Homework**. Construct the Cayley (multiplication) table for S(A) in Homework*. Example 4 (Homework) 3

4 Example 4 Let G be the set of complex numbers given by G 5 {1, 21, i, 2i}, where i 5!21, and consider the operation of multiplication of complex numbers in G. The table in Figure 3.3 shows that G is closed with respect to multiplication. Multiplication in G is associative and commutative, since multiplication has these properties in the set of all complex numbers. We can observe from Figure 3.3 that 1 is the iden-! tity element and that all elements have inverses. Each of 1 and 21 is its own inverse, and i and 2i are inverses of each other. Thus G is an abelian group with respect to multiplication i 2i i 2i i i i i 2i i 2i i 1 21 Example 5 (Homework) Example It is an immediate corollary of Theorem 2.28 that the set Z n , 314, 324, c, 3n of congruence classes modulo n forms an abelian group with respect to addition. Example 6 (Homework) Example 6 Let G 5 {e, a, b, c}withmultiplicationasdefinedbythetableinfigure 3.4. Figure 3.4? e a b c e e a b c a a b c e b b c e a c c e a b From the table, we observe the following: 1. G is closed under this multiplication. 2. e is the identity element. 4

5 3. Each of e and b is its own inverse, and c and a are inverses of each other. 4. This multiplication is commutative. This multiplication is also associative, but we shall not verify it here because it is a laborious task. It follows that G is an abelian group. Example 7 The table C in Figure S 3.5 defines a binary operation p on the set S 5 {A, B, C, D}. Figure 3.5 * A B C D A B C A B B C D B A C A B C D D A B D D From the table, we make the following observations: 1. S is closed under p. 2. C is an identity element. 3. D does not have an inverse since DX 5 C has no solution. Thus S is not a group with respect to p. Definition 3.3 Finite Group, Infinite Group, Order of a Group If a group G has a finite number of elements, G is called a finite group,oragroup of finite order. ThenumberofelementsinG is called the order of G and is denoted by either 0 0 o(g)or0g0.ifg does not have a finite number of elements, G is called an infinite group. 0 0 Example 8 { } has order o( G) = 2. In Example 5, ( ) = n. The set Z of all integers is a group under addition, and this is an exaple of an o Z n infinite group. In Example*, the group G = e,σ Properties of Group Elements Theorem 3.4 Properties of Group Elements Let G be a group with respect to a binary operation that is written as multiplication. a. The identity element e in G is unique. b. For each x [ G, the inverse x 21 in G is unique. c. For each x [ G, (x 21 ) 21 5 x. 5

6 d. Reverse order law. For any x and y in G,(xy) 21 5 y 21 x 21. e. Cancellation laws. If a, x, and y are in G, then either of the equations ax 5 ay or xa 5 ya implies that x 5 y. Remark. Theorem 3.5 Equivalent Conditions for a Group Let G be a nonempty set that is closed under an associative binary operation called multiplication. Then G is a group if and only if the equations ax 5 b and ya 5 b have solutions x and y in G for all choices of a and b in G. 6

7 Remark. In a group G, the associative property can be extended to products involving more than three factors. For example, if a 1,a 2,a 3, and a 4 are elements of G, then applications of condition 2 in Definition 3.1 yield dition 2 in Definition 3.1 yield and 3a 1 (a 2 a 3 )4 a 4 5 3(a 1 a 2 )a 3 4 a 4 (a 1 a 2 )(a 3 a 4 ) 5 3(a 1 a 2 )a 3 4 a 4. Definition 3.6 Product Notation 3 4 Let n be a positive integer, n $ 2. For elements a 1, a 2, c, a n in a group G, the expression a 1 a 2 ca n is defined recursively by a 1 a 2 c ak a k11 5 (a 1 a 2 c ak )a k11 for k $ 1. Theorem 3.7 Generalized Associative Law Let n $ 2 be a positive integer, and let a 1, a 2, c, a n denote elements of a group G. For any positive integer m such that 1 # m, n, (a 1 a 2 c am )(a m11 c an ) 5 a 1 a 2 c an. 7

8 Example 1 M m n ( R) is an abelian group with respect to addidtion. This is an example of another infinite group. Example M m n ( ), M m n ( ), M m n ( k ), M m n ( ) is a group with respect to addition. Example The nonzero elements of M n ( R) do not form a group with respect to multiplication. Example 2 The invertible elements of M n ( ), M n ( ), M n ( k ), M n ( ), M n R G with respect to matrix multiplication. ( ) form a group 8