If $G$ is a group, $x,y, \in G$ and $xy =yx$ then prove that $|xy| = \operatorname{lcm}\{|x|,|y|\}$ given that $|x|$ and $|y|$ are coprime
Relationship between $\operatorname{ord}(ab), \operatorname{ord}(a)$, and $\operatorname{ord}(b)$