product of transpositions

Quote:

Originally Posted by r7iris

Show that everly element of S(n) (n>=2) is a product of transpositions of the form (k k+1).
[Hint:(k k+2) = (k k+1)(k+1 k+2)(k k+1).]

Theorem: For $n\geq 2$ every element of $S_n$ can be expressed as a product of transpositions.

Thus, given any premutation we can write it as a product of transpositions and we will show that each of these transpositions is a product of the form $\prod (k,k+1)$ .

Say we are working in $S_{10}$ (products are taken from right to left).

Consider $(1,3)$ . We can write it as $(1,2)(2,3)(1,2)$ .

Consider $(1,4)$ . We can write it as $(1,2)(2,3)(3,4)(2,3)(1,2)$ .

Consider $(2,6)$ . We can write it as $(2,3)(3,4)(4,5)(5,6)(4,5)(3,4)(2,3)$ .

You get the general idea. So that means everything can be expressed in consecutive form by using the theorem.