# 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 (?)).

## Contents

## Definition

### Algebraic statement

Suppose is a field of characteristic zero and is a natural number. Then, any finite subgroup of the general affine group is conjugate in to a subgroup of .

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

### Geometric statement

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

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

## Related facts

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

### Applications

## 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.