Equality of Cycles

Theorem

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

Let:

$\rho = \begin {bmatrix} a_0 & \cdots & a_{k - 1} \end {bmatrix} \in S_n$

$\sigma = \begin {bmatrix} b_0 & \cdots & b_{k - 1} \end {bmatrix} \in S_n$

be $k$-cycles of $S_n$.

For $d \in \Z$, by Integer is Congruent to Integer less than Modulus we can associate to $d$ a unique integer $\tilde d \in \set {0, \ldots, k - 1}$ such that:

$d \equiv \tilde d \pmod k$

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

Choose $i, j \in \set {1, \ldots, k}$ such that:

$\ds a_i = \min \set {a_0, \ldots, a_{k - 1} }$

$\ds b_j = \min \set {b_0, \ldots, b_{k - 1} }$

Then:

$\rho = \sigma$

if and only if:

$\forall d \in \Z: a_{i + d} = b_{j + d}$

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

Proof

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