# Linearly extensible automorphism

From Groupprops

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This article defines an automorphism property related to (or which arises in the context of): linear representation theory

View other automorphism properties related to linear representation theory OR View all terminology related to linear representation theory OR View all automorphism properties

This term is related to: Extensible automorphisms problem

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

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

## History

### Origin

The notion arose through attempts to use representation theory in solving the Extensible automorphisms problem.

## Definition

### Symbol-free definition

Given a group and a field, an automorphism of the group is termed **linearly extensible** if it extends to an inner automorphism (of the general linear group)for every faithful finite-dimensional linear representation of the group over the field.

### Definition with symbols

Given a group and a field an automorphism of is termed **linearly extensible** if for any faithful linear representation , there exists such that for any :

## Relation with other properties

### Stronger properties (subject to further conditions)

- Inner automorphism (unconditionally)
- Finite-extensible automorphism when the field is a prime field and the group is a finite group.
- Class-preserving automorphism when the field is class-determining for the group, which happens to be true in the non-modular case
`For full proof, refer: Class-preserving implies linearly extensible` - Galois-class automorphism in the non-modular case
- Linearly pushforwardable automorphism

### Weaker properties (subject to further conditions)

- Class-preserving automorphism when the field is class-separating (for instance, when the group is finite and the field is a splitting field for the group)
`For full proof, refer: Linearly extensible implies class-preserving` - Galois-class automorphism in the non-modular case