Disjoint Permutations Commute

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Theorem

Let $S_n$ denote the symmetric group on $n$ letters.

Let $\rho, \sigma \in S_n$ such that $\rho$ and $\sigma$ are disjoint.

Then $\rho \sigma = \sigma \rho$.


Proof

Let $\rho$ and $\sigma$ be disjoint permutations.


Let $i \in \Fix \rho$.

Then:

$\map {\sigma \rho} i = \map \sigma i$

whereas:

$\map {\rho \sigma} i = \map \rho {\map \sigma i}$


Aiming for a contradiction, suppose $\map \sigma i \notin \Fix \rho$.

Then because $\sigma$ and $\rho$ are disjoint it follows that:

\(\displaystyle \map \sigma i\) \(\in\) \(\displaystyle \Fix \sigma\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \map {\sigma^2} i\) \(=\) \(\displaystyle \map \sigma i\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \map {\sigma^{-1} \sigma^2} i\) \(=\) \(\displaystyle \map {\sigma^{-1} \sigma} i\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \map \sigma i\) \(=\) \(\displaystyle i\)


But it was previously established that $i \in \Fix \rho$.

This is a contradiction.

Therefore:

$\map \sigma i \in \Fix \rho$

and so:

$\map {\rho \sigma} i = \map \sigma i = \map {\sigma \rho} i$


Let $i \notin \Fix \rho$.

Then:

$i \in \Fix \sigma$

and the same proof can be performed with $\rho$ and $\sigma$ exchanged.

$\blacksquare$


Sources