About the Product of Two Commuting Elements

[Bashyboy]

Homework Statement

I am asked to offer an example of two commuting elements whose product does not have an order equal to the least common multiple of their individual orders.

Homework Equations

The Attempt at a Solution

Consider $-1$ and $1$ in $\mathbb{Z}$. Then $1+(-1) = 0$ which has an order of $1$, but the order of $-1$ and $1$ is infinity.

Would this be an acceptable answer? I find it unsettling for some reason, but I cannot see anything wrong with it.

 

[fresh_42]

This looks a bit like cheating. What is the least common multiple of $\infty$ and $\infty$?
You should consider $( \mathbb{Z_{12}}\, , \,+) = (\mathbb{Z}/12 \mathbb{Z} \, , \, +)$ instead, i.e. the remainders from division by $12$ or simply the little hand on the clock.