# Finite-characteristic-semidirectly extensible implies linearly pushforwardable over prime field

This fact is related to: Extensible automorphisms problem

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## Contents

## Statement

Suppose is a finite group and is a prime not dividing the order of . Let be the prime field with elements. Then, any finite-characteristic-semidirectly extensible automorphism of is linearly pushforwardable over .

## Definitions used

### Finite-characteristic-semidirectly extensible automorphism

`Further information: finite-characteristic-semidirectly extensible automorphism`

An automorphism of is termed **finite-characteristic-semidirectly extensible** if, for any homomorphism where is a finite group, such that is a characteristic subgroup in the semidirect product , the automorphism extends to an automorphism of .

### Linearly pushforwardable automorphism

`Further information: Linearly pushforwardable automorphism`
An automorphism is termed **linearly pushforwardable** for a group over a field if, for any finite-dimensional linear representation , there exists such that for every , we have:

## Facts used

## Proof

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