1. ## m cycle

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

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

2. ## Re: m cycle

$\sigma(a_k) = a_{k+1 (mod\ m)}$, by definition.

what you are being asked to prove, is that $\sigma^2$ takes $a_1\rightarrow a_3 \rightarrow a_5 \dots a_m \rightarrow a_2$

$\sigma^3$ takes $a_1\rightarrow a_4 \rightarrow a_7 \dots a_m \rightarrow a_3$ and so on....