Hall-extensible implies class-preserving

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This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., Hall-extensible automorphism) must also satisfy the second subgroup property (i.e., class-preserving automorphism)
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Statement

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

Definitions used

Hall-extensible automorphism

Further information: Hall-extensible automorphism

An automorphism \sigma of a finite group G is termed Hall-extensible if, for any group H containing G as a Hall subgroup, there is an automorphism \sigma' of H whose restriction to G equals \sigma.

Class-preserving automorphism

Further information: Class-preserving automorphism

An automorphism \sigma of a finite group G is termed class-preserving if it sends every element to a conjugate element.

Related facts

Facts used

  1. Hall-extensible implies Hall-semidirectly extensible
  2. Hall-semidirectly extensible implies linearly pushforwardable over prime field
  3. Linearly pushforwardable implies class-preserving for 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

Proof

Facts (1) and (2) combine to yield that any Hall-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).