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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)
View all automorphism property implications | View all automorphism property non-implications
Get more facts about finite-extensible automorphism|Get more facts about class-preserving automorphism
This fact is related to: Extensible automorphisms problem
View other facts related to Extensible automorphisms problemView terms related to Extensible automorphisms problem |

Contents

Statement

Related facts

Other facts about finite groups proved using the same method

Facts about infinite groups proved using similar constructions

Other results towards the associated conjecture/problem

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

This fact is part of an attempt to prove the finite-extensible automorphisms conjecture, which states that every finite-extensible automorphism of a finite group must be an inner automorphism. The finite-extensible automorphisms conjecture is closely related to the extensible automorphisms conjecture, which makes a similar statement about extensible automorphisms of (possibly infinite) groups. Some related results:

Facts used

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).