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

5.1 Definitions and Notations

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

Example

Multiplication of Permutations That is:

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

.

Example 3:Symmetric group S_4

5.2 Cycle Notation:

Example

Definition Example: Let Write in cycle notation

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

Example

5.3 Properties of Permutations

Example

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:

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

Definition

Examples

Lemma

Theorem 5.5

Example Determine whether the following permutation is even or odd:

Proof: