# 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 Finite-dominating subgroup (?) in $A$: in other words, any finite subgroup of $A$ is conjugate in $A$ to a subgroup of $G$.

## 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 $A$.