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Publish On: 2019-05-24

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 Sn and



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 a1 by b1 and then g replaces b1 by c1 so that ƒg replaces a1 by c1. Similarly, ƒg replaces a2 by c2, a3 by c3, …., an by cn.

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 Sn, ∀ ƒ, g Sn.
 

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