**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 number of transpositions is even, so it is a even permutation.

**Example-2:**

The given permutation is the product of two transposes so it is a even permutation.

**Odd Permutations:**

A permutation is called even 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 a odd permutation.

**Example-2:**

The given permutation is the product of five transposes so it is a 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 are both even.

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

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

Hence, P1 P2 and P2P, are even permutations.

**Case II**. If P1, P2 are both odd. Let Pi and 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 P2P, 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 P, be an odd and P2 be an even permutation. Also let P, and P2 be the product of (2n + 1) and 2 and transpositions respectively, where n and m are positive integers.

Then each of P1 P2 and P2P1 is product of (2n + 1) + 2m i.e. [ 2 ( n+ m )+1] transpositions , where 2(n+ m) + 1 is evidently and 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 and I are odd (See Theorem 2 above),

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