The Extensible Automorphisms Problem is as follows:
Let G be a group, and σ an automorphism of G. Call σ extensible if, for any embedding G ≤ H, there is an automorphism σ' of H such that the restriction of σ' to G is σ. Clearly, any inner automorphism is extensible. Is the converse true? That is, is every extensible automorphism inner?
It turns out that this problem was solved in 1987. See the wiki entry extensible implies inner for more details.
Here are some archives of my past efforts.
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- Linear extensibility: a route to Extensible Automorphisms (Part I)(PDF),(PS): This details a proof of the fact that any extensible automorphism of a finite group must be a class automorphism, using linear representation theory. Many related results are also given.
- Linear extensibility: a route to Extensible Automorphisms (Part II)(PDF),(PS):This explores the notion of linear extensibility further and looks at the "modular representation" case.
- Permutation-extensibility: a route to Extensible Automorphisms(PDF)(PS): This explores the use of permutation representations to prove that any extensible automorphism of a (finite) group must take each subgroup to a conjugate subgroup.
- Automorphism groups, the NPC conjecture and iteratively extensible automorphisms:(PDF),PS This explores further results on extensible automorphisms for groups that occur in relation to automorphism groups of other groups. It also explores the relation with the Potentially characteristic subgroup problem.
- Extensible endomorphisms: This looks at the endomorphism equivalent of the extensible automorphisms problem.
- Extensible endomorphisms of nilpotent groups: This looks at the question: given a constant c and a group of nilpotence class c what are the endomorphisms of this group that can be extended to endomorphisms for any embedding in a group of nilpotence class c?
I have also given two student talks related to the problem:
- Conjugacy-class representation duality and the Extensible Automorphisms Problem
- Semidirect products and the Extensible Automorphisms Problem
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