Show that Sn={(1,2),(1,2,......,n)}

Printable View

- Aug 7th 2008, 02:24 AMmathemanyakPermutation
Show that Sn={(1,2),(1,2,......,n)}

- Aug 7th 2008, 11:01 AMOpalg
If $\displaystyle \sigma = (1,2,\ldots,n)$ then $\displaystyle \sigma^k$ takes j to j+k (mod n) (for j=1,2,...,n). It follows that $\displaystyle \sigma^{k-1}(1,2)\sigma^{-k+1} = (k,k+1)$. But it is well known (see here, for example) that transpositions of adjacent elements generate the whole of S_n.