Order of Product of Disjoint Permutations

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

Let $\pi$ be a product of disjoint permutations of orders $k_1, k_2, \ldots, k_r$.

Then:
 * $\order \pi = \lcm \set {k_1, k_2, \ldots, k_r}$

where:
 * $\order \pi$ denotes the order of $\pi$ in $S_n$
 * $\lcm$ denotes lowest common multiple.

Proof
If $\pi$ is a cycle, then $\order \pi$ is its length because Order of Cycle is Length of Cycle. Since the LCM of a single number is the number itself, the result follows.

Otherwise, let $\pi = \rho_1 \rho_2 \cdots \rho_r$ where each $\rho_s$ is of order $k_s$, and $\rho_1$ to $\rho_r$ are mutually disjoint permutations.

Let $t = \lcm \set {k_1, k_2, \ldots, k_r}$.

From Disjoint Permutations Commute:


 * $\pi^t = \rho_1^t \rho_2^t \cdots \rho_r^t$

We have that:
 * $\forall s: 1 \le s \le r: \exists m \in \Z: t = m s$

Hence:
 * $\forall s: \rho_s^t = e$

Thus $\pi^t = e$ and certainly $\order \pi \divides t$.

But if $\pi^u = e$, then because the disjointedness of $\rho_1, \rho_2, \cdots, \rho_r$ implies the disjointedness of $\rho_1^u, \rho_2^u, \cdots, \rho_r^u$, each $\rho_s^u = e$ so $k_s \divides u$.

Thus, if $\pi^u = e$, then:
 * $t \divides u$

Thus, by choosing $u = \order \pi$:
 * $\order \pi = t = \lcm \set {k_1, k_2, \ldots, k_r}$