Normal join-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 $p$ is said to be normal join-closed or N-closed if it satisfies the following. Whenever there are two normal subgroups of a group, both having the property as abstract groups, then their join also has the property as an abstract group.

Definition with symbols

A group property $p$ is said to be normal join-closed or N-closed if whenever $N_{1},N_{2}\triangleleft G$ such that $N_{1}$ and $N_{2}$ both satisfy $p$ as abstract groups, so does $N_{1}N_{2}$.

Relation with other metaproperties

Stronger metaproperties

• Join-closed group property

Weaker metaproperties

• Direct product-closed group property
• Central product-closed group property