# Characteristic-semidirectly extensible automorphism

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This article defines an automorphism property, viz a property of group automorphisms. Hence, it also defines a function property (property of functions from a group to itself)

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This is a variation of extensible automorphism|Find other variations of extensible automorphism |

This term is related to: Extensible automorphisms problem

View other terms related to Extensible automorphisms problem | View facts related to Extensible automorphisms problem

## Definition

### Definition with symbols

Let be an automorphism of a group . We say that is **characteristic-semidirectly extensible** or a **CSE-automorphism** if the following holds:

Let be a homomorphism such that is a characteristic subgroup of the associated semidirect product . Then, there exists an automorphism of whose restriction to is .

## Relation with other proeprties

### Stronger properties

- Extensible automorphism
- Retraction-extensible automorphism
- Semidirectly extensible automorphism
- Potentially characteristic-semidirectly extensible automorphism

### Weaker properties

For a finite group, linearly pushforwardable automorphism over a prime field where the prime does not divide the order of the group. `For full proof, refer: finite-characteristic-semidirectly extensible implies linearly pushforwardable over prime field`