Subgroup-closed group property

This article defines a group metaproperty: a property that can be evaluated to true/false for any group property
View a complete list of group metaproperties

Definition

Symbol-free definition

A group property is said to be subgroup-closed or S-closed if any subgroup of a group having the property also has the property.

Definition with symbols

A group property $p$ is said to be subgroup-closed or S-closed if whenever $G$ satisfies property $p$ and $H$ is a subgroup of $G$, $H$ must also satisfy property $p$.

Relation with other metaproperties

Stronger metaproperties

• SQ-closed group property
• Quasivarietal group property
• Varietal group property

Weaker metaproperties

• Normal subgroup-closed group property