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Obviously the automorphism groups of finite dimensional vector spaces are studied in great detail. By changing presentations of the vector space (by which I mean isomorphisms to some $K^{\oplus n}$) we can characterize automorphisms as matrices in $GL(n,K)$ and bring them in suitable shapes (thinking of Smith normal form, Jordan normal form etc).

Categorically this might be due to the very special fact that every vector space is free (in particular there are no relations between generators) and that finite products and finite coproducts coincide, making computing automorphisms rather easy.

But other than linear algebra I could not find other fields in mathematics, where automorphism groups are studied in that detail, which might be due to the presence of said relations. In fact it is common to reduce the study of some structure to linear algebra (e.g. via representation theory) to exploit matrices.

So I was wondering, whether there are other categories, in which automorphism groups are of similar use, while being somewhat accessible/computable. Let me give some examples, to clarify what I mean.

  • In Galois theory automorphisms of a field extension $L\mid K$ play a central role. I don’t know how easily they can be computed in specific instances though...
  • Do the classification results of linear algebra carry over to automorphisms of finitely generated abelian groups? It seems like there we can exploit the structure theorem to get at least a block diagonal matrix form. I was unable to find references...
  • Automorphism groups of simple graphs seem to be rather difficult to compute, so it doesn’t seem like there is a general theory of automorphism groups of simple graphs. I would guess there are certain subcategories of good graphs, where automorphisms are better behaved though...
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    $\begingroup$ Are you asking specifically about groups? We can talk about automorphism groups of fields, rings, graphs, designs, geometric objects, etc. In each case, automorphism groups are well-studied, and can be said to be well-understood for many families of examples. Can you be specific about what exactly you are looking for here? $\endgroup$
    – xxxxxxxxx
    Commented Dec 26, 2020 at 14:18
  • $\begingroup$ @MorganRidgers I am curious whether this is a typo or "design" is a name of an actual mathematical object? $\endgroup$
    – lisyarus
    Commented Dec 26, 2020 at 14:52
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    $\begingroup$ Concerning finitely generated abelian groups specifically, you can start with this answer: math.stackexchange.com/questions/55262/… $\endgroup$
    – lisyarus
    Commented Dec 26, 2020 at 15:04
  • $\begingroup$ @MorganRodgers thank, you very much. I think the question was vague indeed, yet I don’t know whether my edit made it more specific... $\endgroup$
    – Jonas Linssen
    Commented Dec 26, 2020 at 16:08
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    $\begingroup$ @lisyarus en.m.wikipedia.org/wiki/Combinatorial_design $\endgroup$
    – Alex Kruckman
    Commented Dec 26, 2020 at 16:39

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Let's say we are talking about groups; though rings and fields, among others, are also interesting.

Due to Cayley's theorem, every group can be embedded in a symmetric group.

For $\mathfrak S_n$ (finite symmetric groups), other than $n=6$, all the automorphisms are inner.

If $n\ne2,6$, then $\rm{Aut}\mathfrak S_n\cong\mathfrak S_n$.

Also, due to the fact that there is only one way to extend an automorphism from $\mathfrak A_n$, the $n$-th alternating group, to $\mathfrak S_n$, we get that $\rm{Aut}\mathfrak S_n\cong\rm{Aut}\mathfrak A_n$.

Next, I would like to mention the automorphisms of finite cyclic groups, $\Bbb Z_n$. They are fairly easy to understand, since an automorphism has to take a generator to a generator. We easily get that $\rm{Aut}\Bbb Z_n\cong\Bbb Z_n^\times$.

The infinite cyclic group, $\Bbb Z$, on the other hand, has just two automorphisms, one the trivial one and one sending $1$ to $-1$.

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    $\begingroup$ So automorphisms of finite symmetric groups are nice, because they are just conjugation. This is lovely. Just out of curiosity: where does the exception 6 come from? $\endgroup$
    – Jonas Linssen
    Commented Dec 26, 2020 at 12:30
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    $\begingroup$ Why does the automorphism group of G have anything to do with automorphism group of a group that contains G? $\endgroup$
    – lisyarus
    Commented Dec 26, 2020 at 14:14
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    $\begingroup$ I agree with lisyarus, I don't see the relevance of the first line of this. $\endgroup$
    – xxxxxxxxx
    Commented Dec 26, 2020 at 14:16
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    $\begingroup$ @ChrisCuster Why this attitude? Downvotes are pretty normal and provide a mechanism to indicate that a post has certain problems. In this case, the sentence on the Cayley theorem seems to implicitly imply that the automorphism of a group has something to do with inner automorphisms of symmetric groups, which I demonstrated not to be the case in my previous comment. If I misunderstood your answer, I will happily retract my downvote after we resolve this misunderstanding. $\endgroup$
    – lisyarus
    Commented Dec 26, 2020 at 14:47
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    $\begingroup$ @PrudiiArca: which informally says that there aren’t enough small numbers to avoid having coincidences happen. $\endgroup$
    – Arturo Magidin
    Commented Dec 26, 2020 at 23:46

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