Show that a permutation with odd order is an even permutatio

Show that a permutation with odd order is an even permutation.

Solution

(i)an odd number of transpositions ---odd permutations and

(ii)an even number of transpositions-- even permutations

There are many ways to write a permutations as the product of transpositions .Products will either and odd or even number of factors but never both.

knowing the parity oddness/eveness can be found easily one can use the following patterns

(a1,a2,a3,a4,a5)=(a1,a5),(a1,a3),(a1,a1),(a1,a2)

which is even because there are four transpositions you will see that the number of transpositions is a product corresponding to the permutations that can be expressed as the product of transpositions i.e. cycle of length n can be expressed as product of (n-1) transpositions

So we can say that the permutation with odd order must be even number of permutations and also the permutation with even order must be even number of permutations