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. 

Another one is $GL(2,\mathbb Q)$. Take $A=\begin{pmatrix}
  0 & -1 \\
 1 & 0
\end{pmatrix}, B=\begin{pmatrix}
  0 & 1 \\
 -1 & -1
\end{pmatrix}$. Try to show that $A^4=B^3=E$ but $AB$ has infinite order. This shows that if the group $G$ is not abelian then $tG$, including the torsion elements of $G$, may not be a subgroup.
