# Why is the order of a $k$-cycle $\sigma$ equal to $k$?

If $\sigma = (a_1,a_2,\ldots,a_k)$, then the order of $\sigma$ is $k$ because $\sigma^k = Id$. I've tried finding a proof on the internet, but all sources just say "it's clear that", etc. I'm probably missing something really obvious. Can anyone tell me why this is?

Edit: I understand the intuition --- what I'm struggling with is a proof, a formal understanding of why this is true.

• Can you write down what $\sigma^2,\,\sigma^3$ do? – Daniel Fischer Feb 12 '14 at 12:31
• Sure, and I intuitively know why $\sigma^k = Id$, but what I'm struggling with is a formal proof or understanding of why this works. – Newb Feb 12 '14 at 12:34
• I'd personally draw a picture. Have The elements $a_1,\ldots, a_k$ at the top and the bottom and draw lines between the $a_i$ at the top and $\sigma(a_i)$ at the bottom. Now repeat this for $k$ rows of connecting intervals. If you follow any path, you will see that you end up back at the start. This is for intuition, not so much a formal proof. – Dan Rust Feb 12 '14 at 12:38
• You prove by induction that $\sigma^n(a_{r}) = a_{r+n}$ for $1 \leqslant r \leqslant k$, where the addition is done modulo $k$. And of course that $\sigma^n(b) = b$ for $b\notin \{ a_r : 1\leqslant r \leqslant k\}$. Then you see that $\sigma^n \neq \operatorname{id}$ for $1 \leqslant n \leqslant k$. – Daniel Fischer Feb 12 '14 at 12:39

Denote $\sigma=(a_0,a_1,\ldots,a_{k-1})$ (to simplify notation later on).
Proposition: for all $n\in\Bbb N$ and $j\in\Bbb Z/k\Bbb Z$ it holds that $\sigma^n(a_j)=a_{j+n\pmod{k}}$.
Proof: by induction, since trivally $\sigma(a_{j+n\pmod{k}})=a_{j+n+1\pmod{k}}$.
Now note that by our proposition $\sigma^k=\mathrm{Id}$, and for all $j<k$ we have $\sigma^j\ne\mathrm{Id}$.