Powers of Disjoint Permutations

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Theorem

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

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


Then:

$\forall k \in \Z: \paren {\sigma \rho}^k = \sigma^k \rho^k$


Proof

A direct application of Power of Product of Commutative Elements in Group, and the fact that Disjoint Permutations Commute.

$\blacksquare$


Sources