A permutation is said to be an even permutation if it can be expressed as a product of an even number of transpositions; otherwise it is said to be an odd permutation, i.e. it has an odd number of transpositions.
Theorem 1: A permutation cannot be both even and odd, i.e. if a permutation is expected as a product of transpositions then the number of transpositions is either always even or always odd.
Proof: Let us consider the polynomial in distinct symbols . It is defined as the product of factor of the form where .
Now consider any permutation on symbol . By we mean the polynomial obtained by permuting the subscript of the as prescribed by .
For example, taking , we have
If , then
In particular if , we have
This shows that the effect of a transposition on is to change the sign of .
In general, a transposition has the following effects on .
(i) Any factor which involves neither the suffix nor remains unchanged.
(ii) The single factor changes its sign.
(iii) The remaining factors which involve either the suffix or but not both can be grouped into pairs of products, where or and such a product remains unaltered when and are interchanged.
Hence the net effect of transposition on is to change its sign, i.e. operated upon by transposition gives .
Now the permutation is considered a product of transposition when operated upon and gives so that and is considered a product of transposition when it gives so that .
Now this equation will hold only if and are either both even or both odd. Hence this completes the theorem.
Theorem 2: Of the permutations on symbols, are even permutations and are odd permutations.
Proof: Let the even permutations be and the odd permutations be . Then
Now let be any transposition. Since is evidently an odd permutation, we see that are odd permutations and that are even permutations. Since an odd permutation is never an even permutation, we have for any ; . Furthermore, if , then by cancellation law. Similarly if .
It follows that all of the even permutations must appear in the list , which are all distinct so that their number is . Similarly, all of the odd permutations must be in the list , which are all distinct as shown above and their number of .
(1) A cyclic containing an odd number of symbols is an even permutation, whereas a cycle containing an even number of symbols is an odd permutation, since a permutation on symbols can be expressed as a product of transpositions.
(2) The inverse of an even permutation is an even permutation and the inverse of an odd permutation is an odd permutation.
(3) The product of two permutations is an even permutation if either both the permutations are even or both are odd and the product is an odd permutation if one permutation is odd and the other even.