# Orthogonal group of quadratic form

Orthogonal group of the quadratic form over fields, somehow, is well-studied. Indeed E. Cartan has proved for quadratic forms over the reals or complexes that any orthogonal transformation is a product of at most $n$ symmetries, where $n$ is the dimensionality of the underlying vector space. This result was generalized by Dieudonne to quadratic forms over arbitrary base fields.

One can try to understand the integral orthogonal group of an integral quadratic form, more precisely:

Question: Let $q(x_1,\dots,x_n)$ be an integral quadratic form, can we say the integral orthogonal group, denoted as usual by $O_\mathbb{Z}(q)$, is finitely generated. If this would be the case, what can we say about the number of generator?

Let me pick the following special example: $$q(x,y,z)=x^2+y^2-z^2$$ One can show $O_\mathbb{Z}(q)$ acts transitively on $$\{(x,y,z)\in \mathbb{Z}^3: q(x,y,z)=0\}$$ So understanding the number of generators of $O_\mathbb{Z}(q)$, could gives us the space of integral solutions of $q(x,y,z)=0$. For instance, Keith Conrad has a wonderful note entitled with "Orthogonal group of $x^2 + y^2 - z^2$", who proved $O_\mathbb{Z}(q)$ is generated by five elements (I think it was known before but Conrad exposition is great).

This example shows, that the above question can be interesting. What do we know bout the about question?

• Do you have a reference for the theorem of Dieudonne? – wishcow Oct 25 '12 at 17:11
• @wishcow Pete Clark has a nice exposition of this theorem and other aspects of quadratic forms. Check p22 of math.uga.edu/%7Epete/quadraticforms.pdf – Somatic Custard Dec 7 '17 at 14:36

When the form has rank $>1$, then the group has property (T), and therefore is finitely generated by a result of Kazhdan. The original proof though appears to be due to Borel-Harish Chandra. I think this may also be proved using the Borel-Serre compactification. For information about arithmetic groups, check out the book (in progress) by Dave Witte-Morris. In this case, the groups are generated by reflections up to finite index. Venkataramana has some results on the number of generators of such lattices and finite-index subgroups.