Relation between the general linear group of a field and the one of the valuation field.

Question

Let $K$ be a discrete valuation field and let $\mathcal{O}_K$ be its valuation ring. There is some "structural" relation between $GL_n(K)$ and $GL_n(\mathcal{O}_K)$?

And, there is some "structural" relation between $PGL_n(K)$ and $PGL_n(\mathcal{O}_K)$?

Is it possible to generate the corresponding group on the field with the subgroup on the ring and only one element more?

Answer 1

The question leaves some room for possible "relations", so I will just mention an interesting relationship concerning finite subgroups.

Proposition: If the ring of integers $\mathcal{O}_K$ has class number $1$ (so is a PID), then every finite subgroup $G$ in $GL(n,K)$ is conjugated to a subgroup of $ GL(n,\mathcal{O}_K)$.

In particular, every finite subgroup of $GL(n,\mathbb{Q})$ can be conjugated to a subgroup of $GL(n,\mathbb{Z})$.