Any linear group is finite-dominating in the corresponding affine group over characteristic zero

Statement
Suppose $$k$$ is a field of characteristic zero and $$G$$ is a subgroup of the general linear group $$GL(n,k)$$. Consider the group $$A = k^n \rtimes G$$, i.e., the affine group correpsonding to $$G$$. Then, $$G$$ is a fact about::finite-dominating subgroup in $$A$$: in other words, any finite subgroup of $$A$$ is conjugate in $$A$$ to a subgroup of $$G$$.

Applications

 * Orthogonal group is finite-dominating in affine orthogonal group
 * Special orthogonal group is finite-dominating in affine special orthogonal group

Facts used

 * 1) uses::General linear group is finite-dominating in general affine group over characteristic zero
 * 2) uses::Intersection of finite-dominating subgroup with any subgroup whose product with it is the whole group is finite-dominating in it

Proof
Given: A group $$G \le GL(n,k)$$. $$A$$ is the semidirect product of $$k^n$$ and $$G$$. All the groups $$G, A, GL(n,k)$$ and $$GA(n,k)$$ are viewed here as subgroups of $$GA(n,k)$$.


 * 1) By fact (1), $$GL(n,k)$$ is finite-dominating in $$GA(n,k)$$.
 * 2) Since $$A$$ contains $$k^n$$, the product of $$A$$ and $$GL(n,k)$$ is $$GA(n,k)$$. Also, the intersection of $$GL(n,k)$$ with the group $$A$$ is $$G$$. Thus, by fact (2), $$G$$ is finite-dominating in $$A$$.