Isomorph-normal coprime automorphism-invariant subgroup of group of prime power order

This article describes a property that arises as the conjunction of a subgroup property: isomorph-normal coprime automorphism-invariant subgroup with a group property imposed on the ambient group: group of prime power order
View a complete list of such conjunctions | View a complete list of conjunctions where the group property is imposed on the subgroup

Definition

A subgroup $H$ of a group of prime power order $P$ is termed an isomorph-normal coprime automorphism-invariant subgroup of group of prime power order if it satisfies both the following conditions:

1. $H$ is an isomorph-normal subgroup of $P$ (in particular, it is an isomorph-normal subgroup of group of prime power order): In other words, for any subgroup $K$ of $P$ isomorphic to $H$, $K$ is a normal subgroup of $P$.
2. $H$ is a coprime automorphism-invariant subgroup of $P$ (in particular, it is a coprime automorphism-invariant subgroup of group of prime power order).

Relation with other properties

Stronger properties

• Isomorph-free subgroup of group of prime power order
• Isomorph-normal characteristic subgroup of group of prime power order
• Characteristic subgroup of group of prime power order

Weaker properties

• Fusion system-relatively weakly closed subgroup: For proof of the implication, refer Isomorph-normal coprime automorphism-invariant implies weakly closed for any fusion system and for proof of its strictness (i.e. the reverse implication being false) refer Fusion system-relatively weakly closed not implies isomorph-normal.
• Sylow-relatively weakly closed subgroup
• Coprime automorphism-invariant normal subgroup of group of prime power order, coprime automorphism-invariant normal subgroup
• Coprime automorphism-invariant subgroup of group of prime power order, coprime automorphism-invariant subgroup