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

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    The "group-theoretic" way to see what Robert Israel said is to write $GL(2,\mathbb{Z})$ as an amalgamated product of $D_8$ and $D_{12}$, so that every element of finite order lives in (a conjugate of) $D_8$ or $D_{12}$.2012-03-19

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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 $\left(\begin{array}{cc}1 & 1\\ 0 & 1\end{array}\right)^n=\left(\begin{array}{cc}1 & n\\ 0 & 1\end{array}\right)$

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    Thanks for that. That very succinctly answers my question.2012-03-19