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.