Orthogonal group is finite-dominating in general linear group over any real-closed field

This article gives the statement, and possibly proof, of a particular subgroup or type of subgroup (namely, Orthogonal group (?)) satisfying a particular subgroup property (namely, Finite-dominating subgroup (?)) in a particular group or type of group (namely, General linear group (?)).

Definition

Let $k$ be a real-closed field, for instance, $k=\mathbb{R}$, and $n$ be any natural number. Then, any finite subgroup of the general linear group $GL(n,k)$ is conjugate in $GL(n,k)$ to a subgroup of the orthogonal group $O(n,k)$.

Related facts

• General linear group is finite-dominating in general affine group over characteristic zero