Orthogonal Subgroups : on Modules?

[Bacle]

Hi, All:

I have seen Orthogonal groups defined in relation to a pair (V,q) , where V is
a vector space , and q is a symmetric, bilinear quadratic form. The orthogonal
group associated with (V,q) is then the subgroup of GL(V) (invertible linear
maps L:V-->V ), i.e., invertible matrices ( V assumed finite-dimensional), that
preserve the form q, i.e., L in GL(V): q(v1,v2)=q(L(v1),L(v2)).

** Still** I am reading somewhere about what it seems to be an orthogonal
group, but this time associated with a pair (M,q) , where M is not a vector space,
but instead M is a Z-module. I had never seen orthogonal groups extended to
apply to anything other than vector spaces . Does anyone know if this is correct?
If so, what are the properties of these orthogonal groups?

Thanks.

 

[fresh_42]

The orthogonal groups are usually defined over the reals with $q=1$.
Physicists use the Minkowski metric $q=\operatorname{diag}(-1,1,1,1)$ or equivalently $(1,-1,-1,-1)$ and speak also of the orthogonal group, although $q$ is not non degenerate here.

If we pass to integers as scalar area, the lattice group $SL(2,\mathbb{Z})$ is often considered. But of course, we can as well define the orthogonal group for $\mathbb{Z}-$modules. Since the determinant is $1$ we don't have any difficulties. The properties are less groupwise than on the module level, as we have a discrete area which the groups act on.