4 Powers of an Element; Cyclic Groups

Transcription

1 4 Powers of an Element; Cyclic Groups Notation When considering an abstract group (G, ), we will often simplify notation as follows x y will be expressed as xy (x y) z will be expressed as xyz x (y z) will be expressed as xyz In other words, we will omit the symbol, and use product notation, unless a more appropriate notation is called for (e.g., the symbol + will be used in the case of the group (Z, +)). Also, when associativity allows, we can omit parentheses. 1

2 Powers of an Element Notation Let G be a group, and let x be an element of G. We define powers of x as follows: x 0 = e x n = xxx x }{{} n factors x n = x 1 x 1 x 1 x 1 }{{} n factors 2

3 Powers of an Element Theorem 4.1 Let G be a group and let x G. Let m, n be integers. Then: (i) (ii) (iii) x m x n = x m+n (x n ) 1 = x n (x m ) n = x nm = (x n ) m 3

4 Order of an Element Definitions If G is a group and x G, then x is said to be of finite order if there exists a positive integer n such that x n = e. If such an integer exists, then the smallest positive n such that x n = e is called the order of x and denoted by o(x). If x is not of finite order, then we say that x is of infinite order and write o(x) =. 4

5 Order of an Element Examples Let G = (Z 3, ). Then o(1) = 3, since = 0 Let G = (Z, +). Then o(1) =, since

6 Order of an Element Examples Let G = (Q +, ). Then o(2) =, since Let G = GL(2, R), then o (( )) = 2, since ( ) ( ) ( ) ( ) 0 1 =

7 Greatest Common Divisor Let a and b be integers, not both zero. The largest integer d such that d a and d b is called the greatest common divisor of a and b. The greatest common divisor of a and b is denoted by gcd(a, b). 7

8 Greatest Common Divisor Examples gcd(24, 40) = gcd(32, 100) = gcd(12, 91) = 8

9 Relatively Prime The integers a and b are relatively prime if their greatest common divisor is 1. Examples The integers 12 and 25 are relatively prime, since gcd(12, 25) = 1 The integers 27 and 75 are not relatively prime, since gcd(27, 75) =

10 The Euclidean Algorithm Suppose that a and b are positive integers with a b. Successively applying the division algorithm, we obtain a = b q 0 + r 1, 0 r 1 < b, b = r 1 q 1 + r 2, 0 r 2 < r 1, r 1 = r 2 q 2 + r 3, 0 r 3 < r 2, r 2 = r 3 q 3 + r 4, 0 r 4 < r 3, r n 2 = r n 1 q n 1 + r n, 0 r n < r n 1, r n 1 = r n q n. Theorem: gcd(a, b) = r n, where r n is the last nonzero remainder in the sequence above. 10

11 The Euclidean Algorithm Lemma Let a = bq + r, where a, b, q, and r are integers. Then gcd(a, b) = gcd(b, r). Proof: Assume d a and d b. Therefore d r where r = a bq. This shows that all common divisors of a and b are common divisors of b and r. Next, assume d b and d r. Therefore d a where a = bq + r. This shows that all common divisors of b and r are common divisors of a and b. Therefore the common divisors of a and b are the same as the common divisors of b and r. Therefore, their greatest common divisors are the same. 11

12 The Euclidean Algorithm Examples Find gcd(198, 252) using the Euclidean algorithm Find gcd(414, 662) using the Euclidean algorithm 12

13 Greatest Common Divisor Theorem 4.2 If a and b are integers, not both zero, then there exist integers x and y such that gcd(a, b) = ax + by. Proof: Solving for the last nonzero remainder in the Euclidean Algorithm, we have r n = r n 2 r n 1 q n 1 Using the preceding step of the Euclidean Algorithm, we may express the term r n 1 in terms of r n 2 and r n 3, thereby obtaining r n = r n 2 (r n 3 r n 2 q n 2 ) q n 1 After a sequence of n 1 substitutions, we obtain an expression for gcd(a, b) = r n in terms of a and b. 13

14 Greatest Common Divisor Example: Express gcd(198, 252) = 18 as a linear combination of 252 and

15 Greatest Common Divisor Theorem 4.3 If a bc and gcd(a, b) = 1 and, then a c. Proof: Since gcd(a, b) = 1, there exist integers x and y such that ax + by = 1. Multiplying on both sides by c yields axc + byc = c a(xc) + bc(y) = c Since a divides each term on the left-hand side, a also divides the sum of the two terms. Therefore, a divides c. 15

16 Powers of an Element Theorem 4.4 Let G be a group and let x G. Let m, n, d be integers. Then: (i) o(x) = o(x 1 ) (ii) (iii) If o(x) = n and x m = e, then n m If o(x) = n and gcd(m, n) = d, then o(x m ) = n/d. 16

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20 Cyclic Groups Definition A group G is called cyclic if there is an element x G such that G = x = {x n n Z}. The element x is called a generator for G, and the cyclic group generated by x is denoted by x. 17

21 Cyclic Groups Example The group G = (Z 1, ) is the trivial group {0} consisting of one (identity) element. It is the cyclic group generated by x = 0: (Z 1, ) = 0 For all n 2, the group G = (Z n, ) is a finite cyclic group generated by x = 1: (Z n, ) = 1 = {0, 1, 1 1,..., } 1 1 1{{ 1} } n 1 terms 18

22 Cyclic Groups Example The group G = (Z, +) is an infinite cyclic group generated by x = 1: (Z, +) = 1 = {..., ( 1) + ( 1), ( 1), 0, 1, 1 + 1, ,...} The group G = (Q, +) is not cyclic, since there does not exist q Q such that ever rational number r Q has the form r = nq for some n Z. 19

23 Cyclic Groups Theorem 4.5 Let G = x. If o(x) =, then x j x k for j k, and consequently G is infinite. If o(x) = n, then x j = x k iff j k (mod n), and consequently the distinct elements of G are e, x, x 2,..., x n 1. Proof: 20

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25 Cyclic Groups Definition The order of a group G, denoted by G, is the number of elements in G. Corollary 4.6 If G = x, then G = o(x). 21

26 Cyclic Groups Theorem 4.7 If G is a cyclic group, then G is abelian. Proof: 22