Izable subgroup property

This article defines a subgroup metaproperty: a property that can be evaluated to true/false for any subgroup property
View a complete list of subgroup metaproperties
View subgroup properties satisfying this metaproperty| View subgroup properties dissatisfying this metaproperty
VIEW RELATED: subgroup metaproperty satisfactions| subgroup metaproperty dissatisfactions

Definition

Definition with symbols

A subgroup property $p$ is termed izable if for any subgroup $H$ of a group $G$, there is a unique subgroup $M$ of $G$ such that all these conditions are satisfied:

• $M$ contains $H$
• Let $L$ be a subgroup of $G$ containing $H$. Then $H$ satisfies $p$ in $L$ if and only if $L$ is contained in $M$.

Equivalently, a subgroup property is izable if it is upper join-closed, identity-true and satisfies the intermediate subgroup condition.

Relation with other metaproperties

Weaker metaproperties

• Upper join-closed subgroup property
• Identity-true subgroup property
• Intermediate subgroup condition