[Math] Prove that the order of an element a of $S_n$ is the least common multiple of the lengths

Question

Prove that the order of an element a of $S_n$ is the least common multiple of the lengths of the cycles which are obtained when a is written as a product of disjoint cyclic permutations.

Answer 1

Hint:

If $\theta=\sigma_1\sigma_2\cdots\sigma_n$. Where $\sigma_i$ and $\sigma_j$ are disjoint cycles when $i\neq j$. And let $a_1,a_2,\cdots,a_n$ be the orders of $\sigma_1,\cdots,\sigma_n$. Let L.C.M of orders be $t$.

Disjoint cycles commute. Prove using induction that $\theta^k=\sigma_1^k\cdots\sigma_n^k$. Observe that $\theta^t=e$. Hence $o(\theta)\mid t$.

Prove that $\sigma_i^{o(\theta)}=e$ therefore $a_i\mid o(\theta)$. Show that $t\mid o(\theta)$.