Pronormality satisfies intermediate subgroup condition

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

Statement with symbols

Suppose $H\le K\le G$ are groups such that $H$ is a pronormal subgroup of $G$. Then, $H$ is also a pronormal subgroup of $K$.

Related facts

Related metaproperty dissatisfactions for pronormality

• Pronormality does not satisfy transfer condition: We can have a pronormal subgroup $H$ of $G$ and a subgroup $K$ of $G$ such that $H\cap K$ is not pronormal in $K$.
• Pronormality is not upper join-closed: If $H$ is pronormal in intermediate subgroups $K_1,K_2$, it is not necessary that $H$ is pronormal in $⟨K_1,K_2$.

Related properties satisfying the intermediate subgroup condition

• Weak pronormality satisfies intermediate subgroup condition
• Paranormality satisfies intermediate subgroup condition
• Polynormality satisfies intermediate subgroup condition
• Weak normality satisfies intermediate subgroup condition

Proof

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