Equality of Cycles
Jump to navigation
Jump to search
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:
- $\rho = \sigma$
- $\forall d \in \Z: a_{i + d} = b_{j + d}$
That is, $\rho = \sigma$ if and only if they are identical when written with the lowest element first.
Proof
This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |