Fusion system-relatively weakly closed subgroup

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The article defines a subgroup property, where the definition may be in terms of a particular prime number that serves as parameter
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Definition

Suppose $G$ is a group of prime power order, i.e., a finite $p$-group for some prime number $p$. A subgroup $H$ of $G$ is termed a fusion system-relatively weakly closed subgroup if $H$ is a weakly closed subgroup for any fusion system ${\mathcal {F}}$ on $G$.

Relation with other properties

Stronger properties

• Isomorph-normal coprime automorphism-invariant subgroup of group of prime power order: For full proof, refer: Isomorph-normal coprime automorphism-invariant implies weakly closed for any fusion system
  • Isomorph-normal characteristic subgroup of group of prime power order
  • Coprime automorphism-invariant maximal subgroup of group of prime power order
  • Characteristic maximal subgroup of group of prime power order
  • Isomorph-free subgroup of group of prime power order
• Fusion system-relatively strongly closed subgroup
  • Subisomorph-containing subgroup of group of prime power order

Weaker properties

• Sylow-relatively weakly closed subgroup