How do I work out the generator for $GL(2,\Bbb{R})$?

Question

Where $G$ is a group $$G=GL(2,\mathbb{R})$$

$$x=\begin{bmatrix} \cos(\pi/4) & \sin(\pi/4)\\ -\sin(\pi/4)& \cos(\pi/4) \end{bmatrix}$$

How do I work out the generator $\langle x\rangle$... Quite new to group theory so any help would help, thanks.

Answer 1

Your matrix $x$ represents a rotation in $\mathbb{R}^2$ around the origin by the angle $\frac{\pi}4$ clockwise.

In general $R(\phi) = \begin{bmatrix} \cos\phi & \sin\phi \\ -\sin\phi & \cos\phi\end{bmatrix}$ does the same thing with the angle $\phi$.

Clearly if you rotate by $\phi$ and then by $\psi$, it is the same as rotating by $\phi + \psi$. Hence $R(\phi)R(\psi) = R(\phi + \psi)$. You can also check this algebraically, of course.

Also $R(\phi)^{-1} = R(-\phi)$ as the inverse of a rotation by $\phi$ is a rotation by $-\phi$.

Therefore

$$\langle x\rangle = \{x^n : n \in \mathbb{Z}\} = \left\{R\left(\frac{\pi}4\right)^n : n \in \mathbb{Z}\right\} = \left\{R\left(\frac{n\pi}4\right) : n \in \mathbb{Z}\right\}$$

Hence your group consists of all rotations by an angle which is a multiple of $\frac\pi4$.