IMC 1996 · Problem 3

Let $G$ be the subgroup of $GL_2(\mathbb{R})$, generated by $A$ and $B$, where

$$A = \begin{bmatrix} 2 & 0 \\ 0 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}.$$

Let $H$ consist of those 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.

Remarks. $GL_2(\mathbb{R})$ denotes, as usual, the group (under matrix multiplication) of all $2 \times 2$ invertible matrices with real entries (elements). Abelian means commutative. A group is finitely generated if there are a finite number of elements of the group such that every other element of the group can be obtained from these elements using the group operation.