When the automorphism group of an object determines the object

Question

Let me start with three examples to illustrate my question (probably vague; I apologize in advance).

1. $\mathbf{Man}$, the category of closed (compact without boundary) topological manifold. For any $M, N\in \mathbf{Man}$ there is the following theorem (Whittaker) which says that

$$\mathrm{Homeo}(M)\cong_{\text{as groups}} \mathrm{Homeo}(N) \, \textit{ if and only if } \, M\cong_{\text{as manifold}} N$$

2. $\mathbf{NFields}$, the category of Number fields. For any $F, K\in \mathbf{NFields}$ there is a theorem (Neukirch-Uchida) which says that

$$\mathrm{Gal}(\overline{K}/K)\cong_{\text{as progroups}} \mathrm{Gal}(\overline{F}/F) \textit{ if and only if } K\cong_{\text{as fields}} F $$

3. $\mathbf{Vect}$, the category of finite dimensional vector spaces. For any $V, W\in \mathbf{Vect}$ we have that

$$\mathrm{GL}(V)\cong \mathrm{GL}(W) \textit{ if and only if } V\cong W $$

In the first and the third examples we see that the automorphism group of an object determines the object. The second example seems to be similar in some sense however it does not admit the same naive interpretation.

$\textbf{Question:}$ What are other non-trivial examples of interesting categories where the automorphism group of an object determines the object itself? Is there a name for such categories? Is there a way to compare and characterize these kind of categories?

Answer 1

The following result holds.

Theorem.

(1) $\,$ (Baer-Kaplanski) $\,$ If $G$ and $H$ are torsion groups with isomorphic endomorphism rings $\mathrm{End}(G)$ and $\mathrm{End}(H)$, then $G$ and $H$ are isomorphic, and any ring isomorphism $\psi \colon \mathrm{End}(G) \to \mathrm{End}(H)$ is induced by some group isomorphism $\varphi \colon G \to H$.

(2) $\,$ (Leptin-Liebert) If $G$, $H$ are abelian $p$-groups $(p >3)$ and $\mathrm{Aut}(G)$ is isomorphic to $\mathrm{Aut}(H)$, then $G$ is isomorphic to $H$.

See A. V. Mikhalev, G. Pilz: The Concise Handbook of Algebra, p.74 and the references given therein.

Answer 2

If $M$ and $N$ are smooth manifolds, then their diffeomorphisms groups are isomorphic if and only if $M$ and $N$ are diffeomorphic. This smooth counterpart to Whittaker's theorem has been proved by Filipkiewicz (Ergodic Theory and Dynamical Systems, 1982).

Answer 3

If $M,N$ are two countable, $\omega$-categorical and $\omega$-stable structures, and $\operatorname{Aut}(M)\cong \operatorname{Aut}(N)$ (as topological groups), then $M$ and $N$ are bi-interpretable.

More generally, $M$ and $N$ only need to be countable, $\omega$-categorical and satisfy the so-called small index property. As far as I can recall, an analogous result is true for metric structures (in continuous logic).

Answer 4

Under the Generalized Continuum Hypothesis, $$2^{\aleph_\alpha}=\aleph_{\alpha+1}\quad(\forall\alpha),$$ sets with no structure (so automorphisms are just bijections) is an example.

Namely, by

Cardinality of the permutations of an infinite set

we have $\text{card Aut}(X)=2^{\text{card}(X)}$, and $$2^{\aleph_\alpha}=2^{\aleph_\beta}\implies\aleph_{\alpha+1}=\aleph_{\beta+1}\implies\alpha+1=\beta+1\implies\alpha=\beta.$$

(I suppose it's easy to check that we need more than ZFC but do not need GCH here.)

Answer 5

Geometric Complexity Theory:

Theorem (Mulmuley and Sohoni [MS]) The permanent (respectively the determinant) polynomial is characterized by its symmetry group.

That is if $P$ is a homogeneous polynomial of degree $m$ in $m^2$ variables and its symmetry group $G_P$ also fixes the permanent (respectively the determinant), then $P$ must be a scalar multiple of the permanent (respectively the determinant).

Landsberg and Ressayre [LR] made progress on Valiant's version of P vs NP using this result.