Order-dominating subgroup

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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 $K\leq G$ such that the order of $K$ divides the order of $G$, there exists $g\in G$ such that $gKg^{-1}\leq H$.

Relation with other properties

Stronger properties

• Sylow subgroup (in a finite group)
• Hall subgroup in a finite solvable group

Weaker properties

• Order-conjugate subgroup