Intermediate subgroup condition equals upper intersection-closed
This article gives a proof/explanation of the equivalence of multiple definitions for the term intermediate subgroup condition
View a complete list of pages giving proofs of equivalence of definitions
The following are equivalent for a subgroup property :
- satisfies the intermediate subgroup condition: whenever a subgroup of a group satisfies property in , also satisfies property in any intermediate subgroup .
- is closed under finite upper intersections: If and are intermediate subgroups of containing , such that satisfies property in both and , then satisfies property in .
- is closed under arbitrary upper intersections.