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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Further information: Class-separating field
For every finite-dimensional linear representation , and are conjugate in
Then, and are conjugate in .
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 survey articles
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.