CHAPTER 4 - PERMUTATION GROUP.pdf
- 1. ALGEBRAIC STRUCTURES SMA 3033 SEMESTER 1 2023/2024 CHAPTER 4 : PERMUTATION GROUP BY: ASSOC. PROF. DR ROHAIDAH HJ MASRI 1 Chapter 4 SMA3033 Sem 1 2023/2024
- 2. 4.1 PERMUTATION GROUPS Definition 1 (Permutation) A permutation of a set A is a bijective function f : A → A. 1-1 function Onto function This definition preserves the informal idea of rearrangement & has advantage of being applicable to infinite set. We will concentrate on finite set only 2 Chapter 4 SMA3033 Sem 1 2023/2024
- 3. 4.1 PERMUTATION GROUPS Example 1 Let T = { 1, 2, 3 }. The permutation f whose rule is: f(1) = 2, f(2) = 3, f(3) = 1 may represented by the array .1 .2 .3 .1 .2 .3 T T 3 Chapter 4 SMA3033 Sem 1 2023/2024
- 4. Chapter 4 SMA3033 Sem 1 2023/2024 4 4.1 PERMUTATION GROUPS Example 2 Let T = { 1, 2, 3, 4 }. Given f(1) = 4, f(3) = 2, f(2) = 1 and f(4) = 2. Determine whether f is a permutation. Give your reason. Solution: Then, this mapping is not a permutation since 2 appears twice while 3 doesn’t appear at all at the right column.
- 5. Chapter 4 SMA3033 Sem 1 2023/2024 5 4.1 PERMUTATION GROUPS Notes: i. The function composition o is a binary operation on the collection of all permutation of a set A. This operation is called permutation multiplication. ii. If f & g are permutations, then the composite function (g o f ) is defined by A → A → A gives a mapping of A into A. iii. ( g o f ) on A must be read in right-to-left order. f g
- 6. Chapter 4 SMA3033 Sem 1 2023/2024 6 4.1 PERMUTATION GROUPS Example 3 Suppose that A = {1, 2, 3}. Let f & g are the permutations given by: Determine ( f o g ). Solution: ( f o g ) is the function given by: ( f o g )(1) = f(g(1)) = f(2) = 2 ( f o g )(2) = ( f o g )(3) = f(g(2)) = f(1) = 3 f(g(3)) = f(3) = 1 Multiplying in right-to-left order
- 7. Chapter 4 SMA3033 Sem 1 2023/2024 7 4.1 PERMUTATION GROUPS Also can be represented by, ( f o g ) = Example 4 By using functions f & g in example 3, find ( g o f ) and show that ( g o f ) ( f o g ). Answer: Then, (g o f ) ( f o g ).
- 8. Chapter 4 SMA3033 Sem 1 2023/2024 8 4.1 PERMUTATION GROUPS Example 5 Let A be a set and let f and g be permutations of A. Prove that the composite function ( g o f ) is a permutation. Proof: ( To show ( g o f ) is bijective ) Let a, b A. ( g o f )(a) = ( g o f )(b) g(f (a)) = g(f (b)) f (a) = f (b) a = b. Then, ( g o f ) is 1-1. f 1-1 g 1-1
- 9. Chapter 4 SMA3033 Sem 1 2023/2024 9 4.1 PERMUTATION GROUPS ( To show (g o f) is onto ) Let yA. (To show exist xA such that ( g o f )(x) = y.) hypothesis : f onto For all zA , xA such that f(x) = z g onto For all yA , zA such that g(z) = y (g o f)(x) = g(f(x)) = g(z) = y Since g is onto Since f is onto Then, (g o f) is onto. Therefore, (g o f) is a permutation.
- 10. Chapter 4 SMA3033 Sem 1 2023/2024 10 4.1 PERMUTATION GROUPS Example 6 Let G be a group. Prove that the function a : G → G where a(x) = xa for aG and xG, is a permutation. (Try proof this. ) Definition 2 (Symmetric Group) Let A be the finite set { 1, 2, 3, …, n }. The group of all permutations of A is the symmetric group on n letters, and denoted by Sn .
- 11. Chapter 4 SMA3033 Sem 1 2023/2024 11 4.1 PERMUTATION GROUPS Note: The order (the number if element) of Sn is n! . n! = n ( n – 1)(n – 2) … (3)(2)(1) Example 7 Let A = {1, 2, 3} and S3 is a group of 3! = 6 elements. The six permutations of A are:
- 12. Chapter 4 SMA3033 Sem 1 2023/2024 12 4.1 PERMUTATION GROUPS Notes: 1. Let B = {1, 2, 3, 4, 5}. Then S5 is a symmetric group of 5! = 125. 2. If n 3, then Sn is non abelian. (see example 3 & example 4) 3. For a permutation of f, the inverse function f-1 is the permutation that reverses the direction of mapping f .
- 13. Chapter 4 SMA3033 Sem 1 2023/2024 13 4.1 PERMUTATION GROUPS Example 8 Let A = { 1, 2, 3, 4, 5, 6 }. Give three permutations in S6 : Find: i. f -1 ii. f -2 g iii. f -1 g2 h
- 14. Chapter 4 SMA3033 Sem 1 2023/2024 14 4.1 PERMUTATION GROUPS Answer: i. ii. iii.
- 15. Chapter 4 SMA3033 Sem 1 2023/2024 15 4.1 PERMUTATION GROUPS Definition 3 (Cycle) Let a1, a2, …, ak (with k 1) be distinct elements of the set { 1, 2, 3, …, n }. Then, ( a1, a2, …, ak ) denotes the permutation in Sn that maps: a1 to a2 , a2 to a3 , … , ak – 1 to ak , and ak to a1 and maps every other element of {1, 2, 3, …, n } to itself. Here, ( a1, a2, …, ak ) is called a cycle of length k or k-cycle.
- 16. Chapter 4 SMA3033 Sem 1 2023/2024 16 4.1 PERMUTATION GROUPS Example 9 (a) In S4, (1 4 3) is a cycle of length 3 that maps: 1 to 4 4 to 3 3 to 1 2 to 2 1 4 3 It was written in the old notation. Note that: (1 4 3) = (4 3 1) = (3 1 4)
- 17. Chapter 4 SMA3033 Sem 1 2023/2024 17 4.1 PERMUTATION GROUPS Example 9 (b) Write permutation in S5 as a cycle. Solution: = (1 3 5 4) = (3 5 4 1) = (5 4 1 3) = (4 1 3 5)
- 18. Chapter 4 SMA3033 Sem 1 2023/2024 18 4.1 PERMUTATION GROUPS Example 9 (c) (Try!) Express the following permutations of {1, 2, 3, 4, 5, 6, 7} as a cycle. i. ii. Disjoint cycles Two cycles are said to be disjoint if they have no elements in common. Example: (1 3)o (2 5 4 6) represents disjoint cycles but, (1 3) & (3 4 5) are not disjoint cycles since 3 appears in both cycles.
- 19. Chapter 4 SMA3033 Sem 1 2023/2024 19 4.1 PERMUTATION GROUPS Example 10 Express as a product of disjoint cycles. i. ii. iii. (1 4)o (2 3 5 7)o (1 3 4 7 2)
- 20. Chapter 4 SMA3033 Sem 1 2023/2024 20 4.1 PERMUTATION GROUPS Theorem 1 If = ( a1 a2 … ak ) and = ( b1 b2 … br ) are disjoint cycles in Sn , then o = o . Theorem 2 Every permutation of a finite set is a product of disjoint cycles.
- 21. Corollary 1 Any permutation of a finite set of at least two elements is a product of transposition. Chapter 4 SMA3033 Sem 1 2023/2024 21 4.1 PERMUTATION GROUPS Even & Odd Permutations Definition 4 (Transposition) A cycle of length 2 is a transposition. Note: ( a1 a2 …. an ) = ( a1 an ) o ( a1 an-1 ) o … o (a1 a3 ) o ( a1 a2 )
- 22. Chapter 4 SMA3033 Sem 1 2023/2024 22 4.1 PERMUTATION GROUPS Example 11 Write each permutation as a product of transposition. i. ii. iii. = (1 2) o (4 5) o (6 7 9) = (1 3 6 5 4 2) = (1 2) o (1 4) o (1 5) o (1 6) o (1 3) = (1 4 7 6 2 8 3) = (1 3) o (1 8) o (1 2) o (1 6) o (1 7) o (1 4)
- 23. Chapter 4 SMA3033 Sem 1 2023/2024 23 4.1 PERMUTATION GROUPS Definition 5 (Even/ Odd Permutation) A permutation in Sn is said to be even if it can be written as the product of an even number of transpositions, and odd if it can be written as the product of an odd number of transpositions. Example 12 Determine whether each permutation in example 11 is even or odd.
- 24. Chapter 4 SMA3033 Sem 1 2023/2024 24 4.1 PERMUTATION GROUPS Theorem 2 No permutation in Sn can be written as the product of an even number of transpositions and also as the product of an odd transpositions. Definition 6 (Alternating Group) The set of all even permutations in Sn is called the alternating group and denoted by An .
- 25. Chapter 4 SMA3033 Sem 1 2023/2024 25 4.1 PERMUTATION GROUPS Example 13 Determine whether the following permutations are even or odd. i. (1 2 3) i. (1 3 2 4 ) = (1 3) o (1 2) even = (1 4) o (1 2) o (1 3) odd Theorem 3 (Cayley’s Theorem) Every group is isomorphic to a group of permutation.