Strongly intersection-closed subgroup property

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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


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.

Relation with other metaproperties

Stronger metaproperties

Weaker metaproperties