Strongly intersection-closed 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
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VIEW RELATED: subgroup metaproperty satisfactions| subgroup metaproperty dissatisfactions

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

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.