Intermediate subgroup condition equals upper intersection-closed

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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

Statement

The following are equivalent for a subgroup property p:

  • p satisfies the intermediate subgroup condition: whenever a subgroup H of a group G satisfies property p in G, H also satisfies property p in any intermediate subgroup K.
  • p is closed under finite upper intersections: If H \le G and K_1, K_2 are intermediate subgroups of G containing H, such that H satisfies property p in both K_1 and K_2, then H satisfies property p in K_1 \cap K_2.
  • p is closed under arbitrary upper intersections.