SQ-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 SQ-closed if it satisfies the following equivalent conditions:

• It is both subgroup-closed and quotient-closed.
• Every subquotient of a group having the property also has the property.

Definition with symbols

A group property $p$ is said to be SQ-closed if it satisfies the following equivalent conditions:

• Whenever $G$ satisfies $p$, every subgroup $H$ of $G$ and every quotient $G/N$ of $G$ also satisfy $p$.
• Whenever $G$ satisfies $p$, $H$ is a subgroup of $G$ and $N$ is a normal subgroup of $H$, then $H/N$ also satisfies $p$.

Relation with other metaproperties

Weaker metaproperties

• Subgroup-closed group property
• Quotient-closed group property