# Pronormality satisfies intermediate subgroup condition

DIRECT: The fact or result stated in this article has a trivial/direct/straightforward proof provided we use the correct definitions of the terms involved
View other results with direct proofs
VIEW FACTS USING THIS: directly | directly or indirectly, upto two steps | directly or indirectly, upto three steps|
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)
View all subgroup metaproperty satisfactions | View all subgroup metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for subgroup properties
Get more facts about pronormal subgroup |Get facts that use property satisfaction of pronormal subgroup | Get facts that use property satisfaction of pronormal subgroup|Get more facts about intermediate subgroup condition

## 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 $\langle K_1, K_2$.

## Proof

This is direct from the definition. PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]