TITLE: What are the possible orders of elements in $GL(2,\mathbb{R})$?
QUESTION [8 upvotes]: This question arose during discussions about interesting examples of "orders of group elements" for a group theory course.
Definition:  $GL(2,\mathbb{R})$ is the group of $2 \times 2$ matrices with real number entries, with non-zero determinant.  The binary operation for this group is matrix multiplication.

Question: What is $\{\mathrm{ord}(M):M \in GL(2,\mathbb{R})\}$?

REPLY [13 votes]: It can be any number. Take a matrix that represents a rotation with an angle $\phi=\frac{2\pi}{n}$ ($n\in \mathbb{Z}$), that is $$\left(\begin{array}{cc}\cos\phi & \sin \phi\\-\sin\phi & \cos\phi\end{array}\right)$$
Its order is clearly $n$.
EDIT: Just for completeness, it can of course also be infinite. An example would be
$$\left(\begin{array}{cc}1 & 1\\ 0 & 1\end{array}\right)$$
which is invertible and satisfies
$$\left(\begin{array}{cc}1 & 1\\ 0 & 1\end{array}\right)^n=\left(\begin{array}{cc}1 & n\\ 0 & 1\end{array}\right)$$