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 $k$.

Related facts

• Maschke's averaging lemma
• Orthogonal group is finite-dominating in general linear group over any real-closed field

Applications

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

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.