Conjugate-permutability satisfies intermediate subgroup condition

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

 * 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 $$HH^g = H^gH$$ 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.