Show that everly element of S(n) (n>=2) is a product of transpositions of the form (k k+1).
[Hintk k+2) = (k k+1)(k+1 k+2)(k k+1).]
Theorem: For every element of 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 .
Say we are working in (products are taken from right to left).
Consider . We can write it as .
Consider . We can write it as .
Consider . We can write it as .
You get the general idea. So that means everything can be expressed in consecutive form by using the theorem.