Even and Odd Permutations and their theorems

Even Permutations:

A permutation is called even if it can be expressed as a product of even number of transpositions.

Example-1:     

Here we can see that the permutation ( 1  2  3 ) has been expressed as a product of transpositions in three ways and in each of them the number of transpositions is even, so it is an even permutation.

Example-2:

The given permutation is the product of two transpositions so it is an even permutation.

Odd Permutations:

A permutation is called odd if it can be expressed as a product of odd number of transpositions.

Example-1:

Here we can see that the permutation ( 3  4  5  6 ) has been expressed as a product of transpositions in two ways and in each of them number of transpositions is odd, so it is an odd permutation.

Example-2:

The given permutation is the product of five transposes so it is an odd permutation.

Theorems on Even and Odd Permutations :

Theorem-1:

If P1 and P2 are permutations, then 

• (a) P1 P2 is even provided P1 and P2 are either both even or both odd.
• (b) P1 P2 is odd provided one of P1 and P2 is odd and the other even.

Proof: (a) 

Case I. If P1, P2 both are even. 

Let P1 and P2 be the product of 2n and 2m transpositions respectively, where n and m are positive integers. 

Then each of P1 P2 and P2 P1 is product of 2n + 2m transpositions, where 2n + 2m is evidently an even integer. 

Hence, P1 P2 and P2 P1 are even permutations. 

Case II. If P1, P2 , both are odd. Let P1 P2 be the product of (2n + 1) and (2m + 1) transpositions respectively, where n and m are positive integers. 

Then each of P1 P2 and P2 P1 is the product (2n + 1) + (2m + 1) i.e., 2 (n + m + 1) transpositions, where 2(n + m + 1) is evidently an even integer. 

Hence, P1 P2 and P2 P1 are even permutations. 

Proof : (b)

Let P1 be an odd and P2 be an even permutation. Also let P1 and P2 be the product of (2n + 1) and 2m transpositions respectively, where n and m are positive integers. 

Then each of P1 P2 and P2 P1 is the product of (2n + 1) + 2m i.e. [ 2 ( n+ m )+1] transpositions , where 2(n+ m) + 1 is evidently an odd integer.

Hence P1 P2 and P2 P1 are odd permutations. 

Theorem-2:
The Identity permutation is an even permutation. 

Proof-: The identity permutation l can always be expressed as the product of two (i.e., even) transpositions. 

For example 

Hence I is an even permutation. (See definition) 

Theorem-3:
The inverse of an even permutation is an even permutation.  

Proof-: If P be an even permutation and P^-1 be its inverse, then PP^-1= I, the identity permutation.

But P and I are even               (See Theorem 2 above), 

so P^-1 is also even                  (See Theorem  1 (a) above)

Theorem-4:
The inverse of an odd permutation is an odd permutation. 

Proof-: If P be an odd permutation and P^-1 be its inverse, then PP^-1= I, the identity permutation.

But P is odd and I is even.            (See Theorem 2 above),

so P^-1 is also odd.                  (See Theorem  1 (b) above)