Hall does not satisfy transfer condition

This article gives the statement, and possibly proof, of a subgroup property (i.e., Hall subgroup) not satisfying a subgroup metaproperty (i.e., transfer condition).
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Statement

It is possible to have a finite group $G$, a Hall subgroup $H$, and a subgroup $K$ of $G$ such that $H\cap K$ is not a Sylow subgroup of $K$.

Related facts

• Sylow does not satisfy transfer condition
• Hall satisfies intermediate subgroup condition

Facts used

1. Hall satisfies transitivity
2. Transitive and transfer condition implies intersection-closed
3. Hall is not intersection-closed

Proof

Property-theoretic proof

The proof follows directly by combining facts (1), (2), and (3).