# Thread: product of transpositions

1. ## product of transpositions

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).]

2. Originally Posted by r7iris
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 $\displaystyle n\geq 2$ every element of $\displaystyle S_n$ 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 $\displaystyle \prod (k,k+1)$.

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

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

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

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

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