# Linearly pushforwardable 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

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This term is related to: Extensible automorphisms problem

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

## Definition

### Definition with symbols

Let be a group and be a field. An automorphism of is termed **linearly pushforwardable** if for any linear representation , there exists such that for any , we have:

## Relation with other properties

### Stronger properties

- Inner automorphism (unconditionally)
- Class-preserving automorphism when the field is a class-determining field (for instance, a field whose characteristic does not divide the order of the group, for a finite group)

### Weaker properties

- Linearly extensible automorphism
- Class-preserving automorphism when the field is a class-separating field (for instance, a splitting field for a finite group)