# Linearly pushforwardable implies class-preserving for class-separating field

This article gives the statement and possibly, proof, of an implication relation between two automorphism properties. That is, it states that every automorphism satisfying the first automorphism property (i.e., linearly pushforwardable automorphism) must also satisfy the second automorphism property (i.e., class-preserving automorphism)

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## Contents

## Statement

In a Class-separating field (?), any Linearly pushforwardable automorphism (?) is class-preserving.

## Definitions used

### Class-separating field

`Further information: Class-separating field`

A field is termed **class-separating** for a group if, given two elements such that:

For every finite-dimensional linear representation , and are conjugate in

Then, and are conjugate in .

### Class-preserving automorphism

`Further information: class-preserving automorphism`

An automorphism of a group is termed class-preserving if it sends every element of the group to an element in its conjugacy class.

### Linearly pushforwardable automorphism

`Further information: Linearly pushforwardable automorphism`

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

## Related facts

## Related survey articles

## Proof

*Given*: A group , a class-separating field for . A linearly extensible automorphism for .

*To prove*: For any , and are conjugate.

*Proof*: Let be any finite-dimensional linear representation of over . Then, since is linearly pushforwardable, the elements and are conjugate inside .

Since this is true for *every* finite-dimensional linear representation , the definition of class-separating field forces us to conclude that and are conjugate.