Let $H$ be an abelian group and $a,b\in H$ with $\mathrm{ord}(a)<\infty$ and $\mathrm{ord}(b)<\infty$.
My question is why $\mathrm{ord}(ab)|(\mathrm{ord}(a)\cdot\mathrm{ord}(b))$ and why there are in a non-abelian group elements with $\mathrm{ord}(ab)\not|(\mathrm{ord}(a)\cdot\mathrm{ord}(b)$ ?
Thank you very much!