Powers of Disjoint Permutations

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

• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $9$: Permutations: Proposition $9.8$