Isomorph-normal coprime automorphism-invariant subgroup of group of prime power order
From Groupprops
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
Contents
Definition
A subgroup of a group of prime power order
is termed an isomorph-normal coprime automorphism-invariant subgroup of group of prime power order if it satisfies both the following conditions:
-
is an isomorph-normal subgroup of
(in particular, it is an isomorph-normal subgroup of group of prime power order): In other words, for any subgroup
of
isomorphic to
,
is a normal subgroup of
.
-
is a coprime automorphism-invariant subgroup of
(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