Orthogonal group for a symmetric bilinear form

This term associates to every field, a corresponding group property. In other words, given a field, every group either has the property with respect to that field or does not have the property with respect to that field

This group property is natural number-parametrized, in other words, for every natural number, we get a corresponding group property

Definition

Let $k$ be a field and $V$ be a vector space over $k$. Consider a symmetric bilinear form $b:V\times V\to k$. Then, the pseudo-orthogonal group for $n$, denoted as $O(b,k)$, is defined as the group of all matrices $A$ such that:

$b(v,w)=b(Av,Aw)$

for all $v,w\in V$.

Facts

Over complex numbers

Over complex numbers, and over any algebraically closed field, any two symmetric bilinear forms are equivalent. Hence, the pseudo-orthogonal groups corresponding to symmetric bilinear forms are all conjugate to the standared orthogonal group.

Over real numbers

Over real numbers, and over any real-closed field, there are $n+1$ types of pseudo-orthogonal groups of order $n$. For each of these, we can choose as the bilinear form, one of the signature matrices (viz a diagonal matrix with some 1s and some -1s).

Over rational numbers

There are lots of inequivalent pseudo-orthogonal groups over the rational numbers.