## The order of an element in Sym(n) is the least common multiple of the signature of its cycle decomposition

Prove that the order of an element in $S_n$ is the least common multiple of the lengths of the cycles in its cycle decomposition.

Let $\sigma \in S_n$ have the form $\sigma = \prod_I \sigma_i$ where the $\sigma_i$ are pairwise disjoint and $\sigma_i$ is an $n_i$-cycle. Let $|\sigma| = m$, and let $\ell = \mathsf{lcm}_I\{n_i\}$. Note that since the $\sigma_i$ are disjoint, we have $\sigma_i^m = 1$; hence $n_i|m$ for each $i \in I$. By the definition of least common multiple, then, $\ell|m$. Now we also have that $\sigma^\ell = \prod_I \sigma_i^\ell = 1$, so that $m|\ell$. Since $m$ and $\ell$ are both positive, $m = \ell$. $\blacksquare$