Assume $(G,\times)$ is a group and for $a,b \in G$: $ab=ba$, $\text{ord}(a)=n$, $\text{ord} (b)=m$
$\operatorname{ord}(u)=r,\,\operatorname{ord}(v)=s$ $\,\Rightarrow\,\operatorname{ord}(uv)=rs\,$ if $\,r,s\,$ (co)primes