# P-automorphism-invariant subgroup

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

## Contents

## Definition

Let be a prime number. A subgroup of a p-group (i.e., a group where the order of every element is a power of ) is termed -automorphism-invariant if, for any automorphism of such that the order of is a power of , .

For the property the context of finite -groups, refer p-automorphism-invariant subgroup of finite p-group.

## Relation with other properties

### Equivalent properties

For a finite -group, the property of being -automorphism invariant equals the property of being a subnormal stability automorphism-invariant subgroup. That's because any -automorphism can be realized as the stability automorphism of a subnormal series, and conversely, any stability automorphism of a subnormal series must be a -automorphism. `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`

### Stronger properties

- Characteristic subgroup:
*For proof of the implication, refer Characteristic implies p-automorphism-invariant and for proof of its strictness (i.e. the reverse implication being false) refer p-automorphism-invariant not implies characteristic*.