Strongly intersection-closed subgroup property

This article defines a subgroup metaproperty: a property that can be evaluated to true/false for any subgroup property
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Definition

Symbol-free definition

A subgroup property is said to be strongly intersection-closed if given any arbitrary (possibly empty) family of subgroups each satisfying the subgroup property in the group, the intersection of all the subgroups again satisfies the property.

Definition with symbols

A subgroup property $p$ is termed strongly intersection-closed if given any group $G$ and any (possibly empty) family of subgroups $H_i$ of $G$ indexed by $i \in I$ such that each $H_i$ satisfies $p$ in $G$, the group $\bigcap_{i \in I} H_i$ also satisfies $p$ in $G$.

In other words, a subgroup property is strongly intersection-closed if it is both intersection-closed and identity-true.

Relation with other metaproperties

Stronger metaproperties

• Strongly UL-intersection-closed subgroup property
• Invariance property: For full proof, refer: Invariance implies strongly intersection-closed

Weaker metaproperties

• Intersection-closed subgroup property
• Finite-intersection-closed subgroup property
• Identity-true subgroup property