1. Chapter 5: Permutation Groups  Definitions and Notations  Cycle Notation  Properties of Permutations

2. 5.1 Definitions and Notations

3. We will study only permutations on a finite set A={1,2,3,……,n}.

4. Example

5. Multiplication of Permutations That is:

6. Let A={1,2,3}. We have 6 permutations on A. They are:

7. .

8. Example 3:Symmetric group S_4

11. 5.2 Cycle Notation:

13. Example

14. Definition Example: Let Write in cycle notation

15. Multiplication of cycles In S_8, let a=(13)(27)(456)(8), b=(1237)(648)(5). Then ab=(13)(27)(456)(8) (1237)(648)(5) =(1732)(48)(56) In array form

16. Example

17. 5.3 Properties of Permutations

25. Example

26. In S_7, there are 7!=5040 elements. We determine all possible orders of these elements. Solution: To do so we write each element in S_7 as a product of disjoint cycles, then take the lcm of the lengths of these cycles. We have the following possibilities:

27. Therefore pssible orders are: 7,6,10,5,12,4,3,2,1

28. Definition

30. Examples

31. Lemma

32. Theorem 5.5

34. Example Determine whether the following permutation is even or odd:

35. Proof: