$\pi$ permutation decomposed in $k$ disjoint cycles of length $n-1, \dots, n_k$. Find the order of $\pi$

Question

I'm currently working in the following excercise:

Suppose $\pi$ is the permutation that can be decomposed in $k$ disjoint cycles of length $n_1, \dots, n_k$. Find the order of $\pi$.

I know how to calculate a permutation order but I'm not sure about the calculation of a permutation of disjoint cycles of length $n_1, \dots, n_k$ order.

Thanks in advance for any hint or help.

Answer 1

Since the order of a $k$-cycle is $k$, you need $\operatorname {lcm}(n_1,\dots,n_k)$.

This is pretty much immediate, since disjoint cycles commute.

Answer 2

Let $\pi=C_1C_2\;...\;C_k,$ where $C_i$ are disjoint cycles of length $n_i \;\forall 1\le i \le k.$ Take any $x_i$ from each $C_i$, then by the pigeon hole principle, we have $$x_i^{n_i}=1\; \forall {1\le i \le k}$$

Let $N$ be the lcm of $n_1,n_2,\;...\;,n_k$. Then $\exists{d_1, \; ...,\;d_k}\in \mathbb N$ such that $N=n_id_i\;\forall 1\le i \le k.$

$C_i$ are disjoint, so they are commutative. Hence $(n_1n_2\;...\;n_k)^N=1$, and thus the order of $\pi$ is $N$, the lcm of $n_1, n_2,\;...\;,n_k.$