6
$\begingroup$

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.

Improve this question
$\endgroup$
2
  • 2
    $\begingroup$ I suppose this is some kind of higher-order generalization: mathoverflow.net/questions/5635/… 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) $\endgroup$
    – Kevin Casto
    Commented Feb 14 at 22:48
  • 1
    $\begingroup$ 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. $\endgroup$
    – Benjamin Steinberg
    Commented Feb 15 at 13:23

1 Answer 1

Reset to default
5
$\begingroup$

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).

Improve this answer
$\endgroup$
4
  • 1
    $\begingroup$ cambridge.org/core/journals/… $\endgroup$
    – Antoine Ducros
    Commented Feb 14 at 22:34
  • 1
    $\begingroup$ 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. $\endgroup$
    – Antoine Ducros
    Commented Feb 14 at 22:39
  • $\begingroup$ 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? $\endgroup$
    – LSpice
    Commented Feb 14 at 22:58
  • 1
    $\begingroup$ 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. $\endgroup$
    – Antoine Ducros
    Commented Feb 15 at 5:32

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .