Finite-extensible implies class-preserving

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
View other facts related to Extensible automorphisms problem | View terms related to Extensible automorphisms problem

Statement

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

Related facts

Other results towards the associated conjecture/problem

Further information: Extensible automorphisms problem, extensible automorphisms conjecture, finite-extensible implies inner

• Finite-extensible implies inner: Stronger results can be used to show that in fact, any finite-extensible automorphism of a group is an inner automorphism.
• Extensible implies subgroup-conjugating
• Finite-extensible implies subgroup-conjugating

Other facts about finite groups proved using the same method

• Finite solvable-extensible implies class-preserving: Essentially, the same proof works, because if the original group is solvable, all the bigger groups constructed are also solvable.
• Finite-quotient-pullbackable implies class-preserving
• Hall-extensible implies class-preserving

Facts about infinite groups proved using similar constructions

• Conjugacy-separable with only finitely many prime divisors of orders of elements implies every extensible automorphism is class-preserving
• Conjugacy-separable and aperiodic implies every extensible automorphism is class-preserving

Facts used

1. Finite-extensible implies semidirectly extensible for representation over finite field of coprime characteristic
2. Semidirectly extensible implies linearly pushforwardable for representation over prime field
3. Linearly pushforwardable implies class-preserving for class-separating field
4. Every finite group admits a sufficiently large finite 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 finite-extensible automorphism is linearly pushforwardable over a (finite) 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).