Order-dominating subgroup

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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof.
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RANDOM SUBGROUP PROPERTY: Conjugacy-closed subgroup: A subgroup such that any two elements of the subgroup that are conjugate in the whole group are conjugate in the subgroup.

Definition

Symbol-free definition

A finite subgroup of a group is termed order-dominating if every other subgroup whose order divides its order, is conjugate to a subgroup contained in it.

Definition with symbols

Let $G$ be a group and $H$ be a finite subgroup. Then, $H$ is termed order-dominating in $G$ if, for any subgroup $K \le G$ such that the order of $K$ divides the order of $G$ , there exists $g \in G$ such that $gKg^{-1} \le H$ .