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 $\rho = \begin{bmatrix} a_0 & \cdots & a_{k-1} \end{bmatrix}$, $\sigma = \begin{bmatrix} b_0  & \cdots & b_{k-1} \end{bmatrix} \in S_n$ be $k$-cycles of $S_n$.

For any $d \in \Z$, by Congruence to an Integer less than Modulus we can associate to $d$ a unique integer $\tilde d \in \left\{0,\ldots,k-1\right\}$ such that $d \equiv \tilde d \mod k$.

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

Choose $i,j \in \left\{ 1,\ldots,k\right\}$ such that:
 * $\displaystyle a_i = \min\left\{ a_0,\ldots,a_{k-1}\right\},\quad b_j = \min\left\{ b_0,\ldots,b_{k-1}\right\}$

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.