Orthogonal group is finite-dominating in affine orthogonal group over characteristic zero

Algebraic statement
Suppose $$k$$ is a field of characteristic zero and $$n$$ is a natural number. $$G$$ is a finite subgroup of the affine orthogonal group $$AO(n,k)$$. Then, $$G$$ is conjugate in $$AO(n,k)$$ to a finite subgroup of the orthogonal group $$O(n,k)$$.

Geometric statement
Suppose $$k$$ is a field of characteristic zero and $$n$$ is a natural number. $$G$$ is a finite subgroup of the affine orthogonal group $$AO(n,k)$$, with the usual action on $$k^n$$. Then, there exists a point of $$k^n$$ fixed under all elements of $$G$$.

When $$k = \R$$, the field of real numbers, this reduces to the statement that every finite subgroup of the fact about::group of Euclidean motions has a fixed point.

Related facts

 * Orthogonal group is finite-dominating in general linear group over any real-closed field
 * General linear group is finite-dominating in general affine group over characteristic zero