Why is the method to finding the order of a torsion subgroup different than finding the maximum order of a given element of a direct product?

Question

If we want to find the maximum order of a given element of say $\mathbb{Z}_3\times \mathbb{Z}_4$. We would have $lcm{(3,4)}=12$ is the maximum order. But if we take a look at finding the order of the torsion subgroup: $$\mathbb{Z}_4 \times \mathbb{Z} \times \mathbb{Z}_3$$ of $$\mathbb{Z}_{12} \times \mathbb{Z} \times \mathbb{Z}_{12}.$$ We would just take $4\times 3=12$ and $12\times 12=144$ for the orders, respectively. Why is it that the when finding the maximum order, you take the least common multiple but when finding the order of a torsion subgroup you just multiply?

Answer 1

The maximum order of an element in a finite group (also called the exponent of a group) is not the same as the order of the group. The two are equal only if the group is cyclic.

In your example $\mathbb Z/4\mathbb Z \times \mathbb Z/4\mathbb Z$ the two are the same only because because the group is cyclic, isomorphic to $\mathbb Z/12\mathbb Z$.

However the Klein 4-group $\mathbb Z/2\mathbb Z \times \mathbb Z/2\mathbb Z$ has order 4 but exponent 2.