Prove that if $\displaystyle \sigma$ is the m cycle $\displaystyle (a_1, a_2,\cdots , a_m)$, then for all $\displaystyle i\in\{1,2,\cdots , m\}$, $\displaystyle \sigma^i(a_k)=a_{k+i}$ where $\displaystyle a_{k+i}$ is replaced by its least positive residue mod m. Deduce that $\displaystyle |\sigma |=m$.

I am not sure about this. If we take $\displaystyle a_k$ unless i is m, we wont get back to k mod m.