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 |
Other results towards the associated conjecture/problem
- 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
- Finite-extensible implies semidirectly extensible for representation over finite field of coprime characteristic
- Semidirectly extensible implies linearly pushforwardable for representation over prime field
- Linearly pushforwardable implies class-preserving for class-separating field
- Every finite group admits a sufficiently large finite prime field
- 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 (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).