Publish On: 2019-03-19

# Ben joy

Total Post: 542

## Question: Describe the product of two permutations

Reply On: 2013-10-07

# Mjay Jollay

Total Post: 0

## ANS: Describe the product of two permutations

The product or composite of two permutations ƒ, g Sn, denoted by

thus the product of two permutations ƒ and g of degree n denoted by ƒg is obtained by first carrying out the operation defined by ƒ and then by g.

Let ƒ, g S

Here the permutation g has been written in such a way that the first row of g coincides with the second row of ƒ. If the product of two permutations ƒ and g is denoted multiplicatively, i.e. by ƒg or ƒog then by definition

For, ƒ replaces a

Obviously, ƒg is also a permutation of degree n. thus the product of two permutations of degree n is also a permutation of degree n.

Therefore, ƒ g S

thus the product of two permutations ƒ and g of degree n denoted by ƒg is obtained by first carrying out the operation defined by ƒ and then by g.

Let ƒ, g S

_{n}andHere the permutation g has been written in such a way that the first row of g coincides with the second row of ƒ. If the product of two permutations ƒ and g is denoted multiplicatively, i.e. by ƒg or ƒog then by definition

For, ƒ replaces a

_{1}by b_{1}and then g replaces b1 by c_{1}so that ƒg replaces a_{1}by c_{1}. Similarly, ƒg replaces a_{2}by c_{2}, a_{3}by c_{3}, …., a_{n}by c_{n}.Obviously, ƒg is also a permutation of degree n. thus the product of two permutations of degree n is also a permutation of degree n.

Therefore, ƒ g S

_{n}, ∀ ƒ, g S_{n}.Like Us On Facebook for All Latest Updates