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 $H\leq K\leq G$, then $H$ is also conjugate-permutable in $K$.

Related facts

• Permutability satisfies intermediate subgroup condition
• Permutability satisfies transfer condition
• Permutability satisfies inverse image condition
• Normality satisfies intermediate subgroup condition
• Normality satisfies transfer condition
• Normality satisfies inverse image condition
• Automorph-permutability does not satisfy intermediate subgroup condition

Applications

• Conjugate-permutable implies subnormal in finite
• Conjugate-permutable implies descendant in slender

Proof

Given: A group $G$, a subgroup $H$ with the property that $HH^{g}=H^{g}H$ for all $g\in G$. $H\leq K\leq G$.

To prove: $HH^{g}=H^{g}H$ for all $g\in K$.

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