# Conjugate-permutability satisfies intermediate subgroup condition

## Contents

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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 $K$.

## 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.