I should specify that $\sigma \in S_n$, the symmetric group. I've written down some permutations and it seems like their order correspond to their lengths. How can I prove this? I was thinking of using induction, but I can't seem to write a cycle with length $r$ into disjoint cycles and thus unable to use the result that disjoint cycles commutes.
Edit: WLOG, I assume $\sigma = (1,2,3,...,r)$. I observed that $\sigma^2:1\mapsto 3$ and $\sigma^3 :1 \mapsto 4$, and so on. It seems quite trivial since every application of $\sigma$ simply maps the current symbol to the next, but I do not know how to write this out properly. Following this logic, then it seems that to have $k$ such that $\sigma^k : x\mapsto x $, $k$ must at least cause it to "loop around", i.e. $k$ has to be at least $r$. But if $k>r$, then it will "overshoot", and so $k=r$. Correct?