Non-Lie Subgroups of GLn(R)

Question

Is there any subgroups of $GL_n(\mathbb{R})$ which are not Lie groups. Some trivial examples I know like $\mathbb{Q}^*\subset \mathbb{R}^*$ which are $0$-dimensional cases. I am looking for some examples where the subgroup has higher dimension. Any help is appreciated.

Answer 1

The matrices of form$\left(\begin{smallmatrix}x&0\\0&q\end{smallmatrix}\right)$, with $x\in\mathbb{R}\setminus\{0\}$ and $q\in\mathbb{Q}\setminus\{0\}$ form a subgroup of $GL_2(\mathbb{R})$ which is not a Lie subgroup.

On the other hand, there's a theorem due to Élie Cartan which states that every closed subgroup of $GL_n(\mathbb{R})$ is again a Lie group.