# Finite-extensible implies class-preserving

From Groupprops

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., finite-extensible automorphism) must also satisfy the second automorphism property (i.e., class-preserving automorphism)

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This fact is related to: Extensible automorphisms problem

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## Statement

Any finite-extensible automorphism of a finite group is class-preserving automorphism.

## Facts used

- Finite-extensible implies Hall-semidirectly extensible
- Hall-semidirectly extensible implies linearly pushforwardable over prime field (where the prime does not divide the order of the group)
- Linearly pushforwardable implies class-preserving when the field is a class-separating field
- Every finite group admits a sufficiently large prime field
- Sufficiently large implies splitting, splitting implies character-separating, character-separating implies class-separating

## Proof

Facts (1) and (2) combine to yield that any finite-extensible automorphism is linearly pushforwardable over a prime field where the prime does not divide the order of the group, and fact (3) yields that if the field chosen is a class-separating field for the group, then the automorphism is class-preserving. Thus, we need to show that for every finite group, there exists a prime field with the prime not dividing the order of the group, such that the field is a class-separating field for the group. This is achieved by facts (4) and (5).