# order of permutation

• March 19th 2008, 06:52 PM
hzhang610
order of permutation
let f=(x_1, x_2,...,x_r) belongs to permutation S_n. show that order of f =r.

How to prove it?
• March 19th 2008, 06:55 PM
ThePerfectHacker
The thing here is to note that if you compose $(x_1,...,x_r)$, $r+1$ times then $x_i$ returns back to its own position. And this happens to be the smallest such number. Thus, $(x_1,...,x_r)^{r+1} = (x_1,...,x_r)$ thus canceling both sides we get $(x_1,...,x_r)^r = \text{Id}$.
• March 19th 2008, 07:41 PM
hzhang610
thanks
You make it looks so easy