Class-preserving implies linearly pushforwardable
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., class-preserving automorphism) must also satisfy the second automorphism property (i.e., linearly pushforwardable automorphism)
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Further information: class-determining field
A field is termed class-determining for a group if, given any two finite-dimensional linear representations of on a vector space over , say , such that for every , the elements and are conjugate inside , we can conclude that and are equivalent.
In other words, the conjugacy classes in of the images of elements in , determine the representation.
Note that what this statement really says is that if two representations are conjugate at every element of , they are equivalent, or globally conjugate.
Further information: Class-preserving automorphism
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 if denotes conjugation by , then . In other words, for any :
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Given: A group , a class-determining field for , a class-preserving automorphism of , and a finite-dimensional linear representation
To prove: There exists such that for every .
Proof: Observe first that and are both linear representations of , since is an automorphism. Further, since is class-preserving, it is true that for any , there exists such that . We thus obtain:
In other words, and are conjugate in .
Now, by the definition of class-determining field, we see that and are equivalent linear representations. Thus, there exists such that for every :