let f=(x_1, x_2,...,x_r) belongs to permutation S_n. show that order of f =r.

How to prove it?

Printable View

- Mar 19th 2008, 06:52 PMhzhang610order 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? - Mar 19th 2008, 06:55 PMThePerfectHacker
The thing here is to note that if you compose $\displaystyle (x_1,...,x_r)$, $\displaystyle r+1$ times then $\displaystyle x_i$ returns back to its own position. And this happens to be the smallest such number. Thus, $\displaystyle (x_1,...,x_r)^{r+1} = (x_1,...,x_r)$ thus canceling both sides we get $\displaystyle (x_1,...,x_r)^r = \text{Id}$.

- Mar 19th 2008, 07:41 PMhzhang610thanks
You make it looks so easy