# Finite-dominating subgroup

From Groupprops

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]

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 of a group is said to be **finite-dominating** if for every finite subgroup of , there exists such that .

## Relation with other properties

### Incomparable properties

## Examples

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

Note that is *not* conjugate-dense.