What does multiplying two permutations do to their order?

Question

when multiplying different permutations together, I noticed that the product will always be of the same order.

Let's suppose these two permutations:

σ₁ = (12)(3) and σ₂ = (132)

That means that σ2σ1 = (32)(1), and σ1σ2 = (13)(2).

Obviously, these products are two different permutations. Yet, they are of the same order and this seems to be true for all products of permutations.

Is there a general rule for the order of a product of two permutations or does anyone have an explanation about what's going on here?

Answer 1

If your permutation $\sigma$ is a product of disjoint cycles $\sigma_1, \dots,\sigma_n$, then the order of the product $\operatorname{ord}(\sigma)=\operatorname{lcm}(\operatorname{ord(\sigma_1),\dots,\operatorname{ord}(\sigma_n)})$, where $\operatorname{lcm}$ is the lowest common multiple.