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# Permutation Groups and Multiplication of Permutation

• Last Updated : 17 Feb, 2021

Permutation: Let G be a non-empty set, then a one-one onto mapping to itself that is is called a permutation.

• The number of elements in finite set G is called the degree of Permutation.
• Let G have n elements then Pn is called set of all permutation of degree n.
• Pn  is also called the Symmetric group of degree n.
• Pn is also denoted by Sn.
• The number of elements in Pn or Sn is Examples:

Case1: Let G={ 1 } element then permutation are Sn or Pn Case 2: Let G= { 1, 2 } elements then permutations are Case 3: Let G={ 1, 2, 3 } elements then permutation are 3!=6. These are, 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: If Find 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,  Attention reader! Don’t stop learning now.  Practice GATE exam well before the actual exam with the subject-wise and overall quizzes available in GATE Test Series Course.

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