Linearly pushforwardable implies class-preserving for class-separating field

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

Definitions used

Class-separating field

Further information: Class-separating field

A field k is termed class-separating for a group G if, given two elements g,h \in G such that:

For every finite-dimensional linear representation \rho:G \to GL(V), \rho(g) and \rho(h) are conjugate in GL(V)

Then, g and h are conjugate in G.

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 \sigma of a group G is termed linearly pushforwardable over a field k if, for any finite-dimensional linear representation \rho:G \to GL(V), there exists an element a \in GL(V) such that for every g \in G, we have:

\rho(\sigma(g)) = a\rho(g)a^{-1}

Related facts

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Given: A group G, a class-separating field k for G. A linearly extensible automorphism \sigma for G.

To prove: For any g \in G, g and \sigma(g) are conjugate.

Proof: Let \rho:G \to GL(V) be any finite-dimensional linear representation of G over k. Then, since \sigma is linearly pushforwardable, the elements \rho(g) and \rho(\sigma(g)) are conjugate inside GL(V).

Since this is true for every finite-dimensional linear representation \rho, the definition of class-separating field forces us to conclude that g and \sigma(g) are conjugate.