Finite-relative-intersection-closed implies transitive

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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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Statement

Suppose p is a subgroup property that is a finite-relative-intersection-closed subgroup property. Explicitly, this means that whenever H,K,L \le G are such that H,K are both contained in L, H satisfies p in G, and K satisfies p in L, then H \cap K satisfies p in G.

Then, p is a transitive subgroup property: if K \le H \le G are groups such that K satisfies p in H and H satisfies p in G, then K satisfies p in G.

Related facts

Proof

We can set L = G with the notation used in the definitions to complete the proof.