On order of elements of a infinite group Let $a$ and $b$ be elements of finite order of an infinite  group $G$. Then do we can say that order $ab$ is finite?
I think this is true for Nilpotent Groups. 

No, we can't. Take for example the modular group. This group can be generated by elements $S^2 = 1$ and $(ST)^3 = 1$. Yet $S(ST) = S^2 T = T$ is a translation which generates (a group isomorphic to) $\mathbb Z$.
