# P-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-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 (i.e., a group of prime power order where the prime is $p$). A subgroup $H$ of $P$ is termed a p-automorphism-invariant subgroup if it satisfies the following equivalent conditions:

1. $H$ is invariant under all the $p$-automorphisms of $P$, where a $p$-automorphism is an automorphism whose order is a power of $p$.
2. $H$ is a subnormal stability automorphism-invariant subgroup of $P$.

## 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
Fully invariant subgroup of group of prime power order

### Weaker properties

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