# $GL_2(\mathbb{Z})$ and the crystallographic restriction theorem

In the comments on this question, Robert Israel proved that the order of an element in $GL_2(\mathbb{Z})$ can be $2,3$ or $6$ (or infinite). This result is remarkedly reminiscent of the crystallographic restriction theorem, but I can't seem to find the relation. Is this mere coincidence, or is there something deep buried here?

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This fact is used in the only prof I've ever seen of the CRT, so yes, there is a connection. –  Alex Becker Mar 19 '12 at 22:52
In the wiki article there are four proofs, but I don't see this fact appearing in none of them. What is the proof you are referring to? –  yohBS Mar 19 '12 at 22:58
It's from Modern Algebra: An Introduction by John Durbin if I remember correctly. I'm afraid I don't have the book on hand. –  Alex Becker Mar 19 '12 at 22:59
Don't forget $1$ and $4$. –  Robert Israel Mar 19 '12 at 23:09
They, of course, follow triviałly. –  yohBS Mar 20 '12 at 5:55

The connection is that any lattice in $\mathbb{R}^2$ is isomorphic to $\mathbb{Z}^2$, so any (linear) group of symmetries of a lattice injects into $\text{GL}_2(\mathbb{Z})$.
Thanks. One has also to add two lines about why it is also valid for lattices in $\mathbb{R}^3$, but it's fairly easy. –  yohBS Mar 20 '12 at 8:58