Strongly intersection-closed subgroup property
From Groupprops
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 is termed strongly intersection-closed if given any group
and any (possibly empty) family of subgroups
of
indexed by
such that each
satisfies
in
, the group
also satisfies
in
.
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