Conjugate-permutability satisfies intermediate subgroup condition

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This article gives the statement, and possibly proof, of a subgroup property (i.e., conjugate-permutable subgroup) satisfying a subgroup metaproperty (i.e., intermediate subgroup condition)
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Statement

Verbal statement

Any conjugate-permutable subgroup of a group is also conjugate-permutable in every intermediate subgroup.

Statement with symbols

If H is a conjugate-permutable subgroup of a group G, and H \le K \le G, then H is also conjugate-permutable in K.

Related facts

Applications

Proof

Given: A group G, a subgroup H with the property that HH^g = H^gH for all g \in G. H \le K \le G.

To prove: HH^g = H^gH for all g \in K.

Proof: Since g \in K and K \le G, we have g \in G, and the statement to prove follows directly from the given data.