# subgroup of $GL_{n+1}\mathbb{R}$ which is isomorphic

Describe a subgroup of $GL_{n+1}\mathbb{R}$ which is isomorphic to the group $\mathbb{R}^n$ under the operation of vector addition.

I have no idea what this would look like. I would really appreciate any help.

• Do it for $n=1$ first. Pick two upper triangular matrices and look at how they multiply. – Mariano Suárez-Álvarez Jan 15 '14 at 7:22

Probably not the expected answer, but how about a group of diagonal matrices with positive diagonal entries? The multiplicative group $(0,+\infty)$ is isomorphic to the additive group $\mathbb{R}$.
• Good one. I guess the author of the problem should have used $\mathbb{Q}$ instead of $\mathbb{R}$ to force the solution that he meant. – Dan Shved Jan 15 '14 at 8:27
Hint. If you bring $\def\R{\Bbb R}\R^n$ in bijection with a hyperplane $H$ in $\R^{n+1}$ that does not pass through the origin (for instance by adding a final coordinate $1$ to vectors), then every translation by a vector in $\R^n$ corresponds to a translation $H\to H$ that can be uniquely extended to a linear operator $\R^{n+1}\to\R^{n+1}$ (which of course must stabilise globally the hyperplane $H$).