How do I prove that an order of a cycle is its length? Let $\sigma$ be a cycle with length $n$ where $\sigma \in S_m$.
How do i prove that $|\langle \sigma \rangle |$ is $n$?
 A: The order of a permutation $\sigma$ is the smallest positive integer $n$ such that $\sigma^n$ is the identity permutation. (This is sometimes a definition, sometimes a consequence of a different definition; I will assume it is known).
Now suppose we have an $n$-cycle $\sigma=(a_0\;a_1\;a_2\;a_3\ldots a_{n-1})$. Its order cannot be less than $n$, because if $0<k<n$ then $\sigma^k(a_0)=a_k$ and it's implicit in the concept of a cycle that $a_0\ne a_k$.
On the other hand, $\sigma^n$ is certainly the identity permutation, because it acts on each element of the cycle by moving it around the entire cycle one time. Formally one can prove (by induction) that $\sigma^k(a_j) = a_{(j+k)\bmod n}$, and since $n\equiv 0\pmod n$ we have $(j+n)\bmod n=j$ whenever $0\le j < n$. And elements that are not in the cycle at all are left alone by $\sigma$ and therefore also by $\sigma^n$.
Since $\sigma^n=e$ and $\sigma^k\ne e$ when $0<k<n$, the order of $\sigma$ is $n$.
(There's no need to divide into odd and even cases as Don Antonio suggests).
A: For example: even length, so that we know that transpositions have order $\;2\;$ and any cycle of length less than $\;2n\;$ has the appropiate order, so that:
$$\sigma=(i_1\,i_2\,\ldots\,i_{2n})\implies \sigma^2=(i_1\,i_3\,\ldots\,i_{2n-1})(i_2\,i_4\,\ldots\,i_{2n})$$
But by the Inductive Hypothesis (IH), we have that
$$\text{ord}(i_1\,i_3\,\ldots\,i_{2n-1})=n=\text{ord}(i_2\,i_4\,\ldots\,i_{2n})$$
and since both $\;n$-cycles are disjoint (and thus they commute), we get
$$\sigma^{2n}=(\sigma^2)^n=(i_1\,i_3\,\ldots\,i_{2n-1})^n(i_2\,i_4\,\ldots\,i_{2n})^n=1$$
But...why can't the order be less than $ \;2n\;$ above? Because that'd mean one of the two $\;n$-cycles above has order less than $\;n\;$ ...
Try now a similar argument for the odd length case (I think it is simpler as you always get, upon exponentiation, again a cycle of the same length...!)
