Finite-relative-intersection-closed implies transitive
This article gives the statement and possibly, proof, of an implication relation between two subgroup metaproperties. That is, it states that every subgroup satisfying the first subgroup metaproperty (i.e., Finite-relative-intersection-closed subgroup property (?)) must also satisfy the second subgroup metaproperty (i.e., Transitive subgroup property (?))
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Suppose is a subgroup property that is a finite-relative-intersection-closed subgroup property. Explicitly, this means that whenever are such that are both contained in , satisfies in , and satisfies in , then satisfies in .
Then, is a transitive subgroup property: if are groups such that satisfies in and satisfies in , then satisfies in .
We can set with the notation used in the definitions to complete the proof.