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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).
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 Hall subgroup|Get more facts about transfer condition

Contents 1 Statement 2 Related facts 3 Facts used 4 Proof 4.1 Hands-on proof 4.2 Property-theoretic proof

Statement

It is possible to have a finite group G, a Hall subgroup H, and a subgroup K of G such that 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

Hands-on proof

Property-theoretic proof

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