Groupprops, The Group Properties Wiki (pre-alpha)
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Contents 1 Definition 2 Relation with other properties 2.1 Stronger properties 2.2 Weaker properties

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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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VIEW FACTS THAT:| prove, or show that, a subgroup satisfies the property
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

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 such that .

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)