# P-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-automorphism-invariant subgroup with a group property imposed on theambient 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 is a prime number and is a finite -group (i.e., a group of prime power order where the prime is ). A subgroup of is termed a **p-automorphism-invariant subgroup** if it satisfies the following equivalent conditions:

- is invariant under all the -automorphisms of , where a -automorphism is an automorphism whose order is a power of .
- is a subnormal stability automorphism-invariant subgroup of .

### Equivalence of definitions

`Further information: Stability group of subnormal series of p-group is p-group, p-group of automorphisms of p-group is contained in stability group of some normal series`

## 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 |