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

From Groupprops

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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 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, so is a group of prime power order. A subgroup of is termed a -core-automorphism-invariant subgroup if satisfies the following equivalent conditions:

- is invariant under , i.e., the p-core of the automorphism group of .
- Every normal -subgroup of sends 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 |