Show that a group does not have a composition series

Question

I am asked to show that the following group does not have a composition series:

$$X = \left\{ \left( \begin{array}{cc} \lambda & 0 \\ \mu & \tau \\ \end{array} \right) : \lambda, \, \mu, \, \tau \in \mathbb{Q}, \, \lambda \neq 0 \neq \tau \right\}$$

We know that if any group is abelian, then it has a composition series if and only if it is finite. But $X$ is not abelian, is it? So I am a bit stuck here. I am not really too sure which direction to take to prove this.I thought maybe to prove that it is isomorphic to a well known group which does not have a composition series, but to be honest I can't think of any.

Another approach I had in this question was to find a normal subgroup of $X$, $N$, such that $N$ does not have a composition series. This would in theory work because if $G$ has a composition series and $M$ is a normal subgroup of $G$ then $M$ has a composition series. However I could not really think of any.

Any sort of help would be very appreciated. Thank you.

Answer 1

The set of matrices $\begin{pmatrix} 1 & 0 \\ \gamma & 1\end{pmatrix}$ forms an infinite abelian normal subgroup.