Results with a flavor “every automorphism of automorphisms is inner”

Question

It seems that there are a number of results which take more or less the following form: let $X$ be some (specific) kind of structure, let $Y$ be the group of automorphisms of $X$ or perhaps ring of endomorphisms something of the sort, and consider an automorphism of $Y$: then it comes from an automorphism of $X$, i.e., it is inner.

Here are some examples which come to my mind:

$\operatorname{Aut}(\mathfrak{S}_n) = \mathfrak{S}_n$ for $n\neq 6$ (here $X$ is a finite set with $n$ elements and $Y$ is $\mathfrak{S}_n$).
The Skolem-Noether theorem that automorphisms of the ring of linear endomorphisms of $k^n$ (with $k$ a field) are inner.
The Dyer-Formanek theorem about the automorphism group of the automorphism group of the free group with rank $n$ (which I was “reminded” of by this question).
We might even argue that the Yoneda lemma has a similar flavor (as it tells us that a homomorphism of homomorphism functors comes from a homomorphism of the underlying category).

I'm sure there are many more, so I think it might be interesting to make a big-list out of them.

I suppose this is some kind of higher-order generalization: https://mathoverflow.net/questions/5635/does-declaremathoperator-autaut-aut-aut-dots-autg-dots-stabilize In particular, see the answer describing how your question is true for $X$ any simple group (which sounds similar to the full statement of Skolem-Noether) — Kevin Casto, Feb 14 '24 at 22:48
Every automorphism of the monoid of all mappings on a set is given by conjugation by a permutation. This result probably goes back to Schreier and has been rediscovered many times. — Benjamin Steinberg, Feb 15 '24 at 13:23

score 5 · Answer 1 · answered Feb 14 '24 at 22:24

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I think there is such a result for the group of diffeomorphisms of a manifold: under some assumptions any iso between Diffeo (M) and Diffeo (N) comes from a diffeo between M and N.

(The assumptions are weak. Compactness maybe, certainly connectedness or pure positive-dimensionality to avoid the set with six elements).

answered Feb 14 '24 at 22:24

Antoine Ducros

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https://www.cambridge.org/core/journals/ergodic-theory-and-dynamical-systems/article/isomorphisms-between-diffeomorphism-groups/BE5563E03AD7DA6A94D08F92CFA4AF90 – Antoine Ducros Feb 14 '24 at 22:34
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When one looks at the article I posted a link to above, one sees that the only assumptions are paracompactness and connectedness. So, almost nothing. – Antoine Ducros Feb 14 '24 at 22:39
Name of the reference: Filipkiewicz - Isomorphisms between diffeomorphism groups. But why tuck these clarifications away in a comment, rather than editing it into your answer? – LSpice Feb 14 '24 at 22:58
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Oh sorry you are right. I behaved like if I were on X rather than on MO. This is too late for editing now, unfortunately. – Antoine Ducros Feb 15 '24 at 05:32

Results with a flavor “every automorphism of automorphisms is inner”

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