Finite-dominating subgroup

From Groupprops
Jump to: navigation, search
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 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.

Relation with other properties

Incomparable properties

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.