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