# Finite-dominating subgroup

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

## Contents

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

## Definition

### Symbol-free definition

A subgroup of a group is said to be finite-dominating if every finite subgroup of the whole group, is conjugate to a finite subgroup within the subgroup.

### Definition with symbols

A subgroup $H$ of a group $G$ is said to be finite-dominating if for every finite subgroup $K$ of $G$, there exists $g \in G$ such that $gKg^{-1} \subset H$.

## Examples

An example is $O(n,\R) \le GL(n,\R)$. Any finite subgroup (and more generally any compact subgroup) of $GL(n,\R)$ can be conjugated to a subgroup inside $O(n,\R)$, by finding an invariant symmetric positive definite bilinear form using the method of averages.

Note that $O(n,\R)$ is not conjugate-dense.