Finite-dominating subgroup

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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

• Conjugate-dense subgroup

Examples

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

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