let f=(x_1, x_2,...,x_r) belongs to permutation S_n. show that order of f =r.
How to prove it?
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 $\displaystyle (x_1,...,x_r)^r = \text{Id}$.