Order-dominated subgroup

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Definition

A subgroup $H$ of a finite group $G$ is termed order-dominated in $G$ if, given any finite subgroup $K$ of $G$ such that the order of $H$ divides the order of $K$, there exists $g\in G$ such that $gH{g}^{-1}\le K$.

Relation with other properties

Stronger properties

• Sylow subgroup: For full proof, refer: Sylow implies order-dominated

Weaker properties

• Order-conjugate subgroup
• Isomorph-conjugate subgroup
• Prehomomorph-dominated subgroup (when the whole group is finite)