Could someone give me a suggestion to solve this problem?
Let $G$ be the subgroup of $GL(2,\mathbb{C})$ defined by
$\left\lbrace \begin{pmatrix} e^{2i\pi t} & 0 \\ 0 & e^{2i\pi\alpha t} \end{pmatrix} : t\in\mathbb{R} \right\rbrace$
Determine the clousure $\overline{G}$ of $G$ in $GL(2,\mathbb{C})$, where $\alpha$ is a irrational number.