Non-Cyclic Subgroup of $GL(2, \mathbb Z)$

Question

I'm trying to find an infinite, non-cyclic, subgroup of:

H =$\left \{ \begin{bmatrix} a & b\\ c & d\end{bmatrix} : a,b,c,d \in \mathbb Z \right\}$

My current thought is to look at the subgroup, $K =\left \{ \begin{bmatrix} a & b \\ c & d \end{bmatrix} : ad - bc = \pm 1 \right \}$.

I feel that this shouldn't be cyclic, but I'm having a lot of trouble proving that. I've been trying proof by contradiction, but that seems to go nowhere.

Any suggestions would be greatly appreciated.

Answer 1

A cyclic group is abelian. To show that your $K$ is not cyclic, therefore, it is enough to exhibit two of its elements which do not commute.

Answer 2

What about orders of elements ? It is not hard to find in your groups elements of finite order like $- I_2$ and elements of infinite order like $\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$. That's impossible in a cyclic group.