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

From Groupprops

## Statement

Suppose is a field of characteristic zero and is a subgroup of the general linear group . Consider the group , i.e., the affine group correpsonding to . Then, is a Finite-dominating subgroup (?) in : in other words, any finite subgroup of is conjugate in to a subgroup of .

## Related facts

### Applications

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

## Facts used

- General linear group is finite-dominating in general affine group over characteristic zero
- 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 . is the semidirect product of and . All the groups and are viewed here as subgroups of .

- By fact (1), is finite-dominating in .
- Since contains , the product of and is . Also, the intersection of with the group is . Thus, by fact (2), is finite-dominating in .