# Permutation Groups and Multiplication of Permutation

Let G be a non-empty set, then a one-one onto mapping to itself that is as shown below is called a **permutation. **

- The number of elements in finite set G is called the degree of Permutation.
- Let G have n elements then P
_{n}is called a set of all permutations of degree n. - P
_{n }is also called the Symmetric group of degree n. - P
_{n}is also denoted by S_{n}. - The number of elements in P
_{n}or S_{n}is

**Examples:**

Case1:Let G={ 1 } element then permutation are S_{n}or P_{n}=

Case 2:Let G= { 1, 2 } elements then permutations are

Case 3:Let G={ 1, 2, 3 } elements then permutation are3!=6.These are,

**Reading the Symbol of Permutation**

Suppose that a permutation is

- First, we see that in a small bracket there are two rows written, these two rows have numbers. The smallest number is 1 and the largest number is 6.
- Starting from the left side of the first row we read as an image of 1 is 2, an image of 1 is 2, an image of 2 is 3, an image of 3 is 1, an image of 4 is 4
**(Self image=identical=identity)**, an image of 5 is 6 and image of 6 is 5. - The above thing can be also read as: Starting from the left side of the first row 1 goes to 2, 2goes to 3, 3goes to,4 goes to 4,5 goes to 6, and 6 goes to 5.

**A cycle of length 2 is called **a **permutation.**

**Example:**

1)Length is 2, so it is a transposition.

2)Length is three, so it is not a transposition.

**Multiplication of Permutation**

Problem: IfFind the product of permutation A.B and B.A

Solution:

Here we can see that in first bracket 1 goes to 2 i.e. image of 1 is 2, and in second row 2 goes to 3 i.e. image of 2 is 3.

Hence, we will write 3 under 1 in the bracket shown below,

Do above step with all elements of first row, answer will be

Similarly,