Prove that the subnormal series $GL_2(\mathbb{R})>N>\{I\}$ cannot be refined to produce a composition series. (Where $N$ is a normal subgroup in $GL_2(\mathbb{R})$). Hence show that $GL_2(\mathbb{R})$ does not have a composition series.
I have thought about going down the lines of a proof by contradiction so we would suppose that $GL_2(\mathbb{R})>N>\{I\}$ can be refined to produce a composition series. So this would mean we could make a refinement such that $$GL_2(\mathbb{R})>H>N>\{I\}$$ or $$GL_2(\mathbb{R})>N>H>\{I\}$$ give us a composition series. But I do not know where to go from here, could anyone help?
Take the cyclic group
$$N:=\left\langle\;\begin{pmatrix}2&0\\0&2\end{pmatrix}\;\right\rangle\,\lhd GL(2,\Bbb R)$$
If $\;GL(2,\Bbb R)\;$ had a composition series then the subnormal series $\;I\lhd N\lhd GL(2,\Bbb R)\;$ could be refined to a comp. series. But this is impossible since $\;N\cong\Bbb Z\;$ and an abelian group has a composition series iff it is finite.