Sylow does not satisfy transfer condition

This article gives the statement, and possibly proof, of a subgroup property (i.e., Sylow subgroup) not satisfying a subgroup metaproperty (i.e., transfer condition).
View all subgroup metaproperty dissatisfactions | View all subgroup metaproperty satisfactions|Get help on looking up metaproperty (dis)satisfactions for subgroup properties
Get more facts about Sylow subgroup|Get more facts about transfer condition|

Statement

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

Related facts

• Hall does not satisfy transfer condition
• Normal Sylow satisfies transfer condition
• Normal Hall satisfies transfer condition

Proof

Property-theoretic proof

We know that the property of being a Sylow subgroup is transitive (a Sylow subgroup of a Sylow subgroup is Sylow). Thus, if the property of being Sylow satisfies the transfer condition, we have that the property of being a Sylow subgroup is intersection-closed, by the general fact Transitive and transfer condition implies finite-intersection-closed.

On the other hand, an intersection of Sylow subgroups need not be Sylow.