# Subgroups of $GL_2(\Bbb R)$ isomorphic to $\Bbb R$

Is this the only subgroup of $$GL_2(\Bbb R)$$ isomorphic to $$\Bbb R$$? $$\left\{\begin{bmatrix}1&a \\ 0&1\end{bmatrix} : a \in \Bbb R\right\}$$ If not, can we describe all such subgroups?

(motivated by this question)

• Hint: Conjugation. Commented Apr 24, 2020 at 15:03
• Up to conjugation, they are all upper triangular, not necessarily strictly. Commented Apr 25, 2020 at 1:16
• You should clarify if you want these subgroups to be isomorphic to ${\mathbb R}$ as topological groups (with subspace topology) or as abstract groups. The answers are different since there are many more subgroups isomorphic to ${\mathbb R}$ as abstract groups. Commented Apr 25, 2020 at 4:13

Well, at the very least there is $$\left\{\begin{bmatrix}1&0 \\ a&1\end{bmatrix} : a \in \Bbb R\right\}$$
• Sorry, I thought that the question is: Find all subgroup of $GL(2)$. Commented Apr 24, 2020 at 16:00
• What about $\left\{\begin{bmatrix}1+a^2&0 \\ a&2+a^2\end{bmatrix} : a \in \Bbb R\right\}$? Commented Apr 24, 2020 at 16:03
• Also $$\left\{\pmatrix{a&0\\0&1}:a>0\right\}?$$ Commented Apr 24, 2020 at 16:08