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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 three 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?

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# 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 three 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?