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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: Fully characteristic subgroup: A subgroup such that any endomorphism of the whole group takes the subgroup to within itself.

Contents 1 Definition 1.1 Symbol-free definition 1.2 Definition with symbols 2 Relation with other properties 2.1 Stronger properties 2.2 Weaker properties

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 such that the order of K divides the order of G, there exists such that .

Relation with other properties

Stronger properties

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

Weaker properties

• Order-conjugate subgroup