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

## Contents

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
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