P-core-automorphism-invariant subgroup of finite p-group

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This article describes a property that arises as the conjunction of a subgroup property: p-core-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

Suppose p is a prime number and P is a finite p-group, so P is a group of prime power order. A subgroup H of P is termed a p-core-automorphism-invariant subgroup if H satisfies the following equivalent conditions:

  1. H is invariant under O_p(\operatorname{Aut}(P)), i.e., the p-core of the automorphism group of P.
  2. Every normal p-subgroup of \operatorname{Aut}(P) sends H to itself.

Relation with other properties

Stronger properties

property quick description proof of implication proof of strictness (reverse implication failure) intermediate notions
Characteristic subgroup of group of prime power order
p-automorphism-invariant subgroup of finite p-group
p-Sylow-automorphism-invariant subgroup of finite p-group

Weaker properties

property quick description proof of implication proof of strictness (reverse implication failure) intermediate notions
Normal subgroup of group of prime power order