Question

The Product of two odd permutation is​

Answer 1

Answer:

the product of two odd permutation is even

Answer 2

Answer:

The product of two odd permutations is an even permutation.

Step-by-step explanation:

Permutation:

• A set S is said to be permutation if it is one -one mapping from S onto itself.

Length of Cycle:  

• Number of elements in any cycle is called the length of cycle

Transposition:

• A cycle of length '2' is called a transposition.  

Odd permutation:

• A permutation is said to be an odd permutation if it can be expressed as the product of odd number of transpositions.  
• Example:
• Let S = (1 2 3 4)
• The above permutation S can be written in form of product of transposition as  (1 4)(1 3)(1 2).
• The permutation S is expressed as product of odd number of transposition.  
• So, S is an odd permutation

 The product of two odd permutations is always an even permutation.

Let us take an example

  Let S = (1 2 3 4) and P = (4 5 6 7) are two odd permutations.

 Product of two permutations S and P is

       = (1 2 3 4)( 5 6 7)

       = (1 4)(1 3)(1 2)(5 7)(5 6)(5 7)

Product has 6 transpositions.

So, Product of two odd permutations has even number of transpositions.

Hence, the product of two odd permutations is an even permutation.

 

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