Finite-extensible implies class-preserving

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

Facts used

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


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).