The order of a permutation is the least positive common multiple of its nontrivial cycle lengths, with value $1$ for the identity

Write $\sigma=\gamma_1\cdots\gamma_r$ as in [L1]. Since the factors commute by [L2], $\sigma^k=\gamma_1^k\cdots\gamma_r^k$ for every natural $k$.

The $k$-th power of a $d_i$-cycle shifts its displayed entries by $k$ positions, so it is the identity exactly when $k\equiv0\pmod{d_i}$, equivalently when $d_i$ divides $k$. Because the supports are disjoint, $\sigma^k$ is the identity exactly when every $\gamma_i^k$ is the identity.

Thus the positive exponents giving the identity are precisely the positive common multiples of $d_1,\ldots,d_r$, so their least element is the order of $\sigma$ by [L3]. If $r=0$, then $\sigma$ is the identity and its order is $1$.