Equality of Cycles

Theorem
Let $S_n$ denote the symmetric group on $n$ letters, realised as the permutations of $\left\{1,\ldots,n\right\}$.

Let $\rho = \begin{bmatrix} a_1 & \cdots & a_k \end{bmatrix}$, $\sigma = \begin{bmatrix} b_1  & \cdots & b_k \end{bmatrix} \in S_n$ be $k$-cycles of $S_n$.

Define $a_d$ and $b_d$ for any $d \in \Z$ by $a_d = a_{d \mod k}$ and $b_d = b_{d \mod k}$

Choose $i,j \in \left\{ 1,\ldots,k\right\}$ such that:
 * $\displaystyle a_i = \min\left\{ a_1,\ldots,a_k\right\},\quad b_j = \min\left\{ b_1,\ldots,b_k\right\}$

Then $\rho = \sigma$ iff for all $d \in \Z$, $a_{i + d} = b_{j + d}$.

That is, $\rho = \sigma$ iff they are identical when written with the lowest element first.