IMC 1996 Problem 9

Let $G$ be the subgroup of $GL_{2}(\mathbb{R})$ generated by $A$ and $B$ , where $$A=\begin{pmatrix} 2 &0 0&1 \end{pmatrix}, B=\begin{pmatrix} 1 &1 0&1 \end{pmatrix}$$ . Let $H$ consist of the matrices $\begin{pmatrix} a_{11} &a_{12} a_{21}& a_{22} \end{pmatrix}$ in $G$ for which $a_{11}=a_{22}=1$ . a) Show that $H$ is an abelian subgroup of $G$ . b) Show that $H$ is not finitely generated.