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

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

## Definition

### Algebraic statement

Suppose $k$ is a field of characteristic zero and $n$ is a natural number. Then, any finite subgroup of the general affine group $GA(n,k)$ is conjugate in $GA(n,k)$ to a subgroup of $GL(n,k)$.

More generally, we can consider $k$ of prime characteristic, but require that the order of the finite subgroup is relatively prime to the characteristic of $k$.

### Geometric statement

Suppose $k$ is a field of characteristic zero and $n$ is a natural number. Consider the natural action of the general affine group $GA(n,k)$ on the vector space $k^n$. Under this action, any finite subgroup of $GA(n,k)$ has a fixed point.

More generally, we can consider $k$ of prime characteristic, but require that the order of the finite subgroup is relatively prime to the characteristic of $k$.

## Proof

### Proof of the geometric statement

The key idea is to take the orbit of any point and consider the average of all the points in that orbit.