Hall-relatively weakly closed subgroup

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Definition

Suppose $G$ is a finite group and $H$ is a subgroup of $G$. We say that $H$ is a Hall-relatively weakly closed subgroup of $G$ if, for any group $L$ containing $G$ as a Hall subgroup, $H$ is a weakly closed subgroup of $G$ relative to $L$.

Relation with other properties

Stronger properties

• Isomorph-free subgroup
• Sylow-relatively weakly closed subgroup: The same definition, but restricted to the case where $G$ is a group of prime power order.

Weaker properties

• Coprime automorphism-invariant normal subgroup: For full proof, refer: Hall-relatively weakly closed implies coprime automorphism-invariant normal
• Coprime automorphism-invariant subgroup
• Normal subgroup