Reconstructing Structures with the Strong Small Index Property up to Bi-Definability

TL;DR

This paper demonstrates that for certain countable structures with specific properties, automorphism groups uniquely determine the structures up to bi-definability, and shows how any finite group can be realized as an outer automorphism group of such structures.

Contribution

It introduces the expanded automorphism group for structures with the strong small index property and no algebraicity, establishing a bi-definability reconstruction result and realizing finite groups as outer automorphism groups.

Findings

Automorphism groups are isomorphic iff the structures are bi-definably equivalent.

Every finite group can be realized as an outer automorphism group of a suitable structure.

Reconstruction up to bi-definability holds for alategorical structures with these properties.

Abstract

Let $\mathbf{K}$ be the class of countable structures $M$ with the strong small index property and locally finite algebraicity, and $\mathbf{K}_*$ the class of $M \in \mathbf{K}$ such that $acl_M(\{ a \}) = \{ a \}$ for every $a \in M$. For homogeneous $M \in \mathbf{K}$, we introduce what we call the expanded group of automorphisms of $M$, and show that it is second-order definable in $Aut(M)$. We use this to prove that for $M, N \in \mathbf{K}_*$, $Aut(M)$ and $Aut(N)$ are isomorphic as abstract groups if and only if $(Aut(M), M)$ and $(Aut(N), N)$ are isomorphic as permutation groups. In particular, we deduce that for $\aleph_0$-categorical structures the combination of strong small index property and no algebraicity implies reconstruction up to bi-definability, in analogy with Rubin's well-known $\forall \exists$-interpretation technique of [7]. Finally, we show that every finite group can be realized as the outer automorphism group of $Aut(M)$ for some countable $\aleph_0$-categorical homogeneous structure $M$ with the strong small index property and no algebraicity.

Equations

$$G_{(K,L)}=\{f\in Aut(M):f\upharpoonright K\in L\}.$$

$$\mathcal{P}\mathcal{S}(M)=\{G_{(K)}:K\in \mathbf{A}(M)\}\text{ and }\mathcal{SS}(M)=\{G_{(K,L)}:(K,L)\in \mathbf{EA}(M)\}.$$

$$\star$$

$$F(\mathbf{j}((Op(f,K))))=Op(F(f),F(\mathbf{j}(K))),$$

$$\begin{array}[]{rcl}F(\mathbf{j}((Op(f,K))))&=&F(fG_{(K)}f^{-1})\\ &=&F(f)F(G_{(K)})(F(f))^{-1}\\ &=&F(f)(F(\mathbf{j}(K))\\ &=&Op(F(f),F(\mathbf{j}(K))),\end{array}$$

$$F(\mathbf{j}((Op(f,(K_1,L_1)))))=Op(F(f),F(\mathbf{j}((K_1,L_1)))),$$

$$\mathcal{M}=\{G_{(a)}:a\in M\}.$$

$$\hat{F}={\mathbf{j}}_N^{-1}F{\mathbf{j}}_M:ExAut(M)\cong ExAut(N).$$

$${\mathbf{j}}_M(P_{\mathbf{A}(M)}^{min})=\mathcal{M}\text{ and }{\mathbf{j}}_N(P_{\mathbf{A}(N)}^{min})=\mathcal{N}.$$

$$F(Aut(M{)}_{(a)})=Aut(N{)}_{f(a)}\in \mathcal{N}.$$

$$\begin{array}[]{rcl}F(h)(f(a))=f(b)&\Leftrightarrow&F(h)Aut(N)_{(f(a))}(F(h))^{-1}=Aut(N)_{(f(b))}\\ &\Leftrightarrow&hAut(M)_{(a)}h^{-1}=Aut(M)_{(b)}\\ &\Leftrightarrow&h(a)=b.\end{array}$$

$$L_{\Gamma }=\{P_{\ell }:\ell <n\}\cup \{R_{\ell ,k}:\ell <k<n\text{ and }\{\ell ,k\}\in E_{\Gamma }\}$$

$$Aut(\Gamma )\cong Aut(Aut(M_{\Gamma }))/Inn(Aut(M_{\Gamma }))=Out(Aut(M_{\Gamma })).$$

Full text

Partially supported by European Research Council grant 338821. No. 1109 on Shelah’s publication list.

1. Introduction

Reconstruction theory deals with the problem of reconstruction of countable structures from their automorphism groups. The first degree of reconstruction that it is usually dealt with is the so-called reconstruction up to bi-interpretability. The second and stronger degree of reconstruction is known as reconstruction up to bi-definability. In group theoretic terms, the first degree of reconstruction corresponds to reconstruction of topological group isomorphisms from isomorphisms of abstract group, while the second degree of reconstruction corresponds to reconstruction of permutation group isomorphisms from isomorphisms of abstract group. Two independent techniques lead the scene in this field: the (strong) small index property (see e.g. [4]) and Rubin’s $\forall\exists$-interpretation [9].

On the reconstruction up to bi-interpretability side the cornerstones of the theory are the following two results:

Theorem** (Rubin [9]).**

Let $M$ and $N$ be countable $\aleph_{0}$-categorical structures and suppose that $M$ has a $\forall\exists$-interpretation. Then $Aut(M)\cong Aut(N)$ if and only if $M$ and $N$ are bi-interpretable.

Theorem** (Lascar [5]).**

Let $M$ and $N$ be countable $\aleph_{0}$-categorical structures and suppose that $M$ has the small index property. Then $Aut(M)\cong Aut(N)$ if and only if $M$ and $N$ are bi-interpretable.

On the reconstruction up to bi-definability side, all the known results are based on the following theorem of Rubin:

Theorem** (Rubin [9]).**

Let $M$ and $N$ be countable $\aleph_{0}$-categorical structures with no algebraicity and suppose that $M$ has a $\forall\exists$-interpretation. Then $Aut(M)\cong Aut(N)$ if and only if $M$ and $N$ are bi-definable.

In particular, on the small index property side there is no result that pairs with the last cited result of Rubin. In this paper we fill this gap proving the following:

Theorem 1**.**

Let $\mathbf{K}_{*}$ be the class of countable structures $M$ satisfying:

1. (1)

$M$* has the strong small index property;* 2. (2)

for every finite $A\subseteq M$, $acl_{M}(A)$ is finite; 3. (3)

for every $a\in M$, $acl_{M}(\{a\})=\{a\}$.

Then for $M,N\in\mathbf{K}_{*}$, $Aut(M)$ and $Aut(N)$ are isomorphic as abstract groups if and only if $(Aut(M),M)$ and $(Aut(N),N)$ are isomorphic as permutation groups. Moreover, if $\pi:Aut(M)\cong Aut(N)$ is an abstract group isomorphism, then there is a bijection $f:M\rightarrow N$ witnessing that $(Aut(M),M)$ and $(Aut(N),N)$ are isomorphic as permutation groups and such that $\pi(\alpha)=f\alpha f^{-1}$.

Thus deducing an analog of Rubin’s result on reconstruction up to bi-definability:

Corollary 2**.**

Let $M$ and $N$ be countable $\aleph_{0}$-categorical structures with the strong small index property and no algebraicity. Then $Aut(M)$ and $Aut(N)$ are isomorphic as abstract groups if and only if $M$ and $N$ are bi-definable. Moreover, if $\pi:Aut(M)\cong Aut(N)$ is an abstract group isomorphism, then there is a bijection $f:M\rightarrow N$ witnessing the bi-definability of $M$ and $N$ such that $\pi(\alpha)=f\alpha f^{-1}$.

For a structure $M$ satisfying the conclusion of Theorem 1 it is easy to determine the outer automorphism group of $Aut(M)$, in fact any $f\in Aut(Aut(M))$ is induced by a permutation of $M$. For example, as already noted by Rubin in [9], using this fact it is easy to see that for $R_{n}$ the $n$-coloured random graph ($n\geqslant 2$) we have that $Out(Aut(R_{n}))\cong Sym(n)$. Similarly, but in a different direction, one easily sees that for $M_{n}$ the $K_{n}$-free random graph ($n\geqslant 3$) we have that $Aut(M_{n})$ is complete. We show here that in this setting any finite group can occur:

Theorem 3**.**

Let $K$ be a finite group. Then there exists a countable $\aleph_{0}$-categorical homogeneous structure $M$ with the strong small index property and no algebraicity such that $K\cong Out(Aut(M))$.

Our main technical tool is what we call the expanded group of automorphisms of an homogeneous structure $M$ with the strong small index property and locally finite algebraicity. This powerful object encodes the combinatorics of $Aut(M)$-stabilizers of such a structure $M$, and it is a crucial ingredient of our proof of Theorem 1. In Theorem 16 we show that the expanded group of automorphisms is second-order orbit-definable in $Aut(M)$ (cf. Definition 15), a fact of essential importance.

The results of this paper pair with those of [8] and [7], where sufficient conditions for strong small index property are isolated and applied in the concrete case of the group of automorphisms of Hall’s universal locally finite group. Finally, we would like to mention another recent result proved by us (see [6]) in this area, i.e. the following powerful non-reconstruction theorem: no algebraic or topological property of $Aut(M)$ can detect any form of stability of the countable structure $M$.

2. The Expanded Group of Automorphisms

In this section we introduce the expanded group of automorphisms of $M$ (for certain $M$), and show that it is second-order definable in $Aut(M)$.

Given a structure $M$ and $A\subseteq M$, and considering $Aut(M)=G$ in its natural action on $M$, we denote the pointwise (resp. setwise) stabilizer of $A$ under this action by $G_{(A)}$ (resp. $G_{\{A\}}$). Also, we denote the subgroup relation by $\leqslant$.

Definition 4**.**

Let $M$ be a structure and $G=Aut(M)$.

1. (1)

We say that $a$ is algebraic (resp. definable) over $A\subseteq M$ in $M$ if the orbit of $a$ under $G_{(A)}$ is finite (resp. trivial). 2. (2)

The algebraic closure of $A\subseteq M$ in $M$, denoted as $acl_{M}(A)$, is the set of elements of $M$ which are algebraic over $A$. 3. (3)

The definable closure of $A\subseteq M$ in $M$, denoted as $dcl_{M}(A)$, is the set of elements of $M$ which are definable over $A$.

Definition 5**.**

Let $M$ be a countable structure and $G=Aut(M)$.

1. (1)

We say that $M$ (or $G$) has the small index property (SIP) if every subgroup of $Aut(M)$ of index less than $2^{\aleph_{0}}$ contains the pointwise stabilizer of a finite set $A\subseteq M$. 2. (2)

We say that $M$ (or $G$) has the strong small index property (SSIP) if every subgroup of $Aut(M)$ of index less than $2^{\aleph_{0}}$ lies between the pointwise and the setwise stabilizer of a finite set $A\subseteq M$.

Hypothesis 6**.**

Throughout this section, let $M$ be a countable homogeneous structure with the strong small index property and locally finite algebraicity, i.e. for every finite $A\subseteq M$ we have $|acl_{M}(A)|<\omega$.

Remark 7**.**

Notice that all $\omega$-categorical structures have locally finite algebraicity.

Notation 8**.**

1. (1)

We let $\mathbf{A}(M)=\{acl_{M}(B):B\subseteq_{fin}M\}$. 2. (2)

We let $\mathbf{EA}(M)=\{(K,L):K\in\mathbf{A}(M)\text{ and }L\leqslant Aut(K)\}$.

Definition 9**.**

Let $(K,L)\in\mathbf{EA}(M)$, we define:

[TABLE]

Notice that if $L=\{id_{K}\}$, then $G_{(K,L)}=G_{(K)}$, i.e. it equals the pointwise stabilizer of $K$, and that if $L=Aut(K)$, then $G_{(K,L)}=G_{\{K\}}$, i.e. it equals the setwise stabilizer of $K$. We then let:

[TABLE]

The crucial point is the following:

Lemma 10**.**

Let $\mathcal{G}=\{H\leqslant G:[G:H]<2^{\omega}\}$. Then $\mathcal{G}=\mathcal{SS}(M)$.

Proof.

The containment from right to left is trivial. Let then $H\leqslant G$ with $[G:H]<2^{\omega}$. By the strong small index property, there is finite $K\subseteq M$ such that $G_{(K)}\leqslant H\leqslant G_{\{K\}}$. It follows that $G_{(acl_{M}(K))}\leqslant H\leqslant G_{\{acl_{M}(K)\}}$, and so without loss of generality we can assume that $K\in\mathbf{A}(M)$. First of all we claim that $G_{(K)}\trianglelefteqslant G_{\{K\}}$. In fact, for $g\in G_{\{K\}}$, $h\in G_{(K)}$ and $a\in K$, we have $ghg^{-1}(a)=gg^{-1}(a)=a$, since $g^{-1}(a)\in K$ and $h\in G_{(K)}$. Furthermore, for $g,h\in G_{\{K\}}$, we have $g^{-1}h\in G_{(K)}$ iff $g\restriction K=h\restriction K$. Hence, the map $f:gG_{(K)}\mapsto g\restriction K$, for $g\in G_{\{K\}}$, is such that:

[TABLE]

since every $f\in Aut(K)$ extends to an automorphism of $M$. Thus, by the fourth isomorphism theorem we have $H=G_{(K,L)}$ for $L=\{f\restriction K:f\in H\}$. ∎

Proposition 11**.**

Let $H_{1},H_{2}\in\mathcal{SS}(M)$. The following conditions are equivalent:

1. (1)

$H_{1}\trianglelefteqslant H_{2}$* and $[H_{2}:H_{1}]<\omega$;* 2. (2)

there is $K\in\mathbf{A}(M)$ and $L_{1}\trianglelefteqslant L_{2}\leqslant Aut(K)$ such that $H_{i}=G_{(K,L_{i})}$ for $i=1,2$.

Proof.

The proof of (2) implies (1) is immediate, since by the normality of $L_{1}$ in $L_{2}$ we have that, for $g\in G_{(K,L_{2})}$ and $h\in G_{(K,L_{1})}$, $ghg^{-1}\restriction K\in L_{1}$, while the fact that $[H_{2}:H_{1}]<\omega$ follows from the proof of Lemma 10. We show that (1) implies (2). By assumption, $H_{i}=G_{(K_{i},L_{i})}$ for $(K_{i},L_{i})\in\mathbf{EA}(M)$ ($i=1,2$).

1. $(*)_{1}$

$K_{2}\subseteq K_{1}$.

Suppose not, and let $a\in K_{2}-K_{1}$ witness this. Then we can find $f\in G$ such that $f\restriction K_{1}=id_{K_{1}}$ and $f(a)\not\in K_{2}$. It follows that $f\in H_{1}-H_{2}$, a contradiction.

1. $(*)_{2}$

$K_{1}\subseteq K_{2}$.

Suppose not, and let $f_{n}\in G$, for $n<\omega$, such that $f_{n}\restriction K_{2}=id_{K_{2}}$, and in addition $\{f_{n}(K_{1}-K_{2}):n<\omega\}$ are pairwise disjoint. Then clearly, for every $n<\omega$, $f_{n}\in H_{2}$ and $\{f_{n}H_{1}:n<\omega\}$ are distinct, contradicting the assumption $[H_{2}:H_{1}]<\omega$.

1. $(*)_{3}$

$L_{1}\leqslant L_{2}$.

Suppose not, and let $h\in L_{1}-L_{2}$. Then $h$ extends to an automorphism $f$ of $M$. Clearly $f\in H_{1}-H_{2}$, a contradiction.

1. $(*)_{4}$

$L_{1}\trianglelefteqslant L_{2}$.

Suppose not, and let $g_{i}\in L_{i}$ ($i=1,2$) be such that $g_{2}g_{1}g_{2}^{-1}\not\in L_{1}$. Then $g_{i}$ extends to an automorphism $f_{i}$ of $M$ ($i=1,2$). Clearly $f_{i}\in H_{i}$ ($i=1,2$), and $f_{2}f_{1}f_{2}^{-1}\not\in H_{1}$, a contradiction. ∎

Proposition 12**.**

Let $\mathcal{G}=\{H\in\mathcal{SS}(M):\text{ there is no }H^{\prime}\in\mathcal{SS}(M),\text{with }H^{\prime}\subsetneq H,H^{\prime}\trianglelefteqslant H\text{ and }[H:H^{\prime}]<\omega\}$. Then $\mathcal{PS}(M)=\mathcal{G}$.

Proof.

First we show the containment from left to right. Let $H_{2}\in\mathcal{PS}(M)$ and assume that there exists $H_{1}\in\mathcal{SS}(M)$ such that $H_{1}\subsetneq H_{2},H_{1}\trianglelefteqslant H_{2}\text{ and }[H_{2}:H_{1}]<\omega$. By Proposition 11, $H_{i}=G_{(K_{i},L_{i})}$ for $(K_{i},L_{i})\in\mathbf{EA}(M)$ ($i=1,2$) and $K_{1}=K=K_{2}$. Now, as $H_{2}\in\mathcal{PS}(M)$, $L_{2}=\{id_{K}\}$. Hence, $L_{1}=L_{2}$, and so $H_{1}=H_{2}$, a contradiction. We now show the containment from right to left. Let $H\in\mathcal{G}$, then $H=G_{(K,L)}$ for $(K,L)\in\mathbf{EA}(M)$. If $L\neq\{id_{K}\}$ then letting $H^{\prime}=G_{(K,\{id_{K}\})}$ we have $H^{\prime}\subsetneq H$, $H^{\prime}\trianglelefteqslant H$ and $[H:H^{\prime}]<\omega$, a contradiction. ∎

Let $\mathbf{L}(M)$ be a set of finite groups such that for every $K\in\mathbf{A}(M)$ there is a unique $L\in\mathbf{L}(M)$ such that $L\cong Aut(K)$.

Proposition 13**.**

Let $L\in\mathbf{L}(M)$ and $H\in\mathcal{SS}(M)$. The following conditions are equivalent:

1. (1)

$H=G_{(K)}\in\mathcal{PS}(M)$* and $Aut(K)\cong L$;* 2. (2)

there is $H^{\prime}\in\mathcal{SS}(M)$ such that $H\trianglelefteqslant H^{\prime}$, $[H^{\prime}:H]<\omega$, $H^{\prime}$ is maximal under these conditions and $H^{\prime}/H\cong L$.

Proof.

Concerning the implication “(1) implies (2)”, let $H^{\prime}=G_{\{K\}}$, then, by Proposition 11 and equation ($\star$ ‣ 2) in the proof of Lemma 10, we have that $H^{\prime}$ is as wanted. Concerning the implication “(2) implies (1)”, if $H$ and $H^{\prime}$ are as in (2), then, by Proposition 11 and equation ($\star$ ‣ 2) in the proof of Lemma 10, it must be the case that $H^{\prime}=G_{\{K\}}$ and $H=G_{(K)}$ for some $K\in\mathbf{A}(M)$ such that $Aut(K)\cong L$. ∎

Definition 14**.**

We define the structure $ExAut(M)$, the expanded group of automorphisms of $M$, as follows:

1. (1)

$ExAut(M)$* is a two-sorted structure;* 2. (2)

the first sort has set of elements $Aut(M)=G$; 3. (3)

the second sort has set of elements $\mathbf{EA}(M)$; 4. (4)

we identify $\{(K,\{id_{K}\}):K\in\mathbf{A}(M)\}$ with $\mathbf{A}(M)$; 5. (5)

the relations are:

1. (a)

$P_{\mathbf{A}(M)}=\{K\in\mathbf{A}(M)\}$* (recalling the above identification);* 2. (b)

for $L\in\mathbf{L}(M)$, $P_{L(M)}=\{K\in\mathbf{A}(M):Aut(K)\cong L\}$; 3. (c)

$\leqslant_{\mathbf{EA}(M)}\;=\{((K_{1},L_{1}),(K_{2},L_{2})):(K_{i},L_{i})\in\mathbf{EA}(M)\;(i=1,2)\text{, }K_{1}\leqslant K_{2}\text{ and }L_{2}\restriction K_{1}\leqslant L_{1}\}$; 4. (d)

$\leqslant_{\mathbf{A}(M)}\;=\{(K_{1},K_{2}):K_{i}\in\mathbf{A}(M)\;(i=1,2)\text{ and }K_{1}\leqslant K_{2}\}$; 5. (e)

$P^{min}_{\mathbf{A}(M)}=\{K\in\mathbf{A}(M):acl(\emptyset)\neq K\in\mathbf{A}(M)\text{ is minimal in }(\mathbf{A}(M),\subseteq)\}$; 6. (6)

the operations are:

1. (f)

composition on $Aut(M)$;

1. (g)

for $f\in Aut(M)$ and $K\in\mathbf{A}(M)$, $Op(f,K)=f(K)$;

1. (h)

for $f\in Aut(M)$ and $(K_{1},L_{1})\in\mathbf{EA}(M)$, $Op(f,(K_{1},L_{1}))=(K_{2},L_{2})$ iff $f(K_{1})=K_{2}$ and $L_{2}=\{f\restriction K_{1}\pi f^{-1}\restriction K_{2}:\pi\in L_{1}\}$.

Definition 15**.**

We say that a set of subsets of a structure $N$ is second-order orbit-definable if it is preserved by automorphisms of $N$. We say that a structure $M$ is second-order orbit-definable in a structure $N$ if there is a injective map $\mathbf{j}$ mapping $\emptyset$-definable subsets of $M$ to second-order orbit-definable set of subsets of $N$.

Theorem 16**.**

Let $M$ and $N$ be as in Hypothesis 6, and let $G=Aut(M)$. Then:

1. (1)

The map $\mathbf{j}_{M}=\mathbf{j}:(f,(K,L))\mapsto(\{f\},G_{(K,L)})$ witnesses second-order orbit-definability of $ExAut(M)$ in $Aut(M)$. 2. (2)

Every $F:Aut(M)\cong Aut(N)$ has an extension $\hat{F}:ExAut(M)\cong ExAut(N)$.

Proof.

We prove (1).

1. $(*)_{1}$

The map $(f,(K,L))\mapsto(\{f\},G_{(K,L)})$ is one-to-one.

Suppose that $(K_{1},L_{1})\neq(K_{2},L_{2})\in\mathbf{EA}(M)$, we want to show that $G_{(K_{1},L_{1})}\neq G_{(K_{2},L_{2})}$. Suppose that $K_{1}\neq K_{2}$. By symmetry, we can assume that $K_{1}\not\subseteq K_{2}$. Then there is $f\in G$ such that $f\restriction K_{2}=id_{K_{2}}$ and $f(K_{1})\neq K_{1}$, since $K_{2}$ is algebraically closed. Thus, $f\in G_{(K_{2},L_{2})}-G_{(K_{1},L_{1})}$. Suppose now that $K_{1}=K=K_{2}$ and $L_{1}\neq L_{2}$. By symmetry, we can assume that $L_{1}\not\subseteq L_{2}$. Let $g\in L_{1}-L_{2}$, then $g$ extends to an automorphism $f$ of $M$. Thus, $f\in G_{(K,L_{1})}-G_{(K,L_{2})}$. Finally, notice that it is not possible that $\{f\}=G_{(K,L)}$, and so we are done.

1. $(*)_{2}$

The range $\mathbf{j}(\mathbf{EA}(M))=\mathcal{SS}(M)$ is mapped onto itself by any $F\in Aut(G)$.

By Lemma 10.

1. $(*)_{3}$

The range $\mathbf{j}(P_{\mathbf{A}(M)}))=\mathcal{PS}(M)$ is mapped onto itself by any $F\in Aut(G)$.

By Proposition 12.

1. $(*)_{4}$

For $L\in\mathbf{L}(M)$, the range $\mathbf{j}(P_{L(M)})=\{G_{(K)}:Aut(K)\cong L\}$ is mapped onto itself by any $F\in Aut(G)$.

By Proposition 13.

1. $(*)_{5}$

The range $\mathbf{j}(\leqslant_{\mathbf{EA}(M)})=\{(G_{(K_{1},L_{1})},G_{(K_{2},L_{2})}):G_{(K_{1},L_{1})}\supseteq G_{(K_{2},L_{2})},(K_{i},L_{i})\in\mathbf{EA}(M)\text{ }(i=1,2)\text{, }K_{1}\leqslant K_{2}\text{ and }L_{2}\restriction K_{1}\leqslant L_{1}\}$ is preserved by any $F\in Aut(G)$.

For $(K_{i},L_{i})\in\mathbf{EA}(M)$ ($i=1,2$) and $F\in Aut(G)$, obviously we have $\mathbf{j}(K_{1},L_{1})\supseteq\mathbf{j}(K_{2},L_{2})$ if and only if $F(\mathbf{j}(K_{1},L_{1}))\supseteq F(\mathbf{j}(K_{2},L_{2}))$, since $F$ induces an automorphism of $(\mathcal{P}(Aut(G)),\subseteq)$.

1. $(*)_{6}$

The range $\mathbf{j}(\leqslant_{\mathbf{A}(M)})=\{(G_{(K_{1})},G_{(K_{2})}):G_{(K_{1})}\supseteq G_{(K_{2})},K_{1},K_{2}\in\mathbf{A}(M),K_{1}\leqslant K_{2}\}$ is preserved by any $F\in Aut(G)$.

As in $(*)_{5}$, i.e. any $F\in Aut(G)$ induces an automorphism of $(\mathcal{P}(Aut(G)),\subseteq)$.

1. $(*)_{7}$

The range $\mathbf{j}(P^{min}_{\mathbf{A}(M)})=\{H\in\mathcal{PS}(M):G\neq H\text{ is maximal in }(\mathcal{PS}(M),\subseteq)\}$ is preserved by any $F\in Aut(G)$.

As in $(*)_{5}$, i.e. any $F\in Aut(G)$ induces an automorphism of $(\mathcal{P}(Aut(G)),\subseteq)$.

1. $(*)_{8}$

For any $F\in Aut(G)$, $F(\{gh\})=F(\{g\})F(\{h\})$.

Obvious.

1. $(*)_{9}$

$\mathbf{j}(Op(f,K))=\mathbf{j}(f(K))=G_{(f(K))}=fG_{(K)}f^{-1}$ and:

[TABLE]

for any $F\in Aut(G)$.

Observe that:

[TABLE]

since by $(*)_{3}$ $\mathcal{PS}(M)$ is mapped onto itself by any $F\in Aut(G)$.

1. $(*)_{10}$

$\mathbf{j}(Op(f,(K_{1},L_{1})))=(fG_{(K_{1})}f^{-1},fG_{(K_{1},L_{1})}f^{-1})$ and:

[TABLE]

for any $F\in Aut(G)$.

Similar to $(*)_{9}$.

This concludes the proof of (1). Finally, (2) follows directly from (1), in fact for $F:Aut(M)\cong Aut(N)$ letting $\hat{F}=\mathbf{j}_{N}^{-1}F\,\mathbf{j}_{M}$ we have $\hat{F}:ExAut(M)\cong ExAut(N)$.

∎

3. Reconstruction and Outer Automorphisms

In this section we prove the theorems stated in the introduction.

Definition 17**.**

Let $\mathbf{K}_{*}$ be the class of countable structures $M$ satisfying:

1. (1)

$M$* has the strong small index property;* 2. (2)

for every finite $A\subseteq M$, $acl_{M}(A)$ is finite; 3. (3)

for every $a\in M$, $acl_{M}(\{a\})=\{a\}$;

As in the previous section, we let $G=Aut(M)$. We denote $G_{(\{a\})}$ simply as $G_{(a)}$. The crucial point in asking this additional condition is the following:

Proposition 18**.**

Let $M\in\mathbf{K}_{*}$ be homogeneous, and define:

[TABLE]

Then $\mathbf{j}_{M}(P^{min}_{\mathbf{A}(M)})=\mathcal{M}$ (recall Definition 14 and Theorem 16).

Proof.

Notice that by the third assumption in Definition 17 we have that $P^{min}_{\mathbf{A}(M)}=\{\{a\}:a\in M\}$, and so directly by the definition of the interpretation $\mathbf{j}_{M}$ (cf. Theorem 16) we have that $\mathbf{j}_{M}(P^{min}_{\mathbf{A}(M)})=\mathcal{M}$. ∎

We will use the suggestive notation $\mathcal{M}=\{G_{(a)}:a\in M\}$ also below.

Definition 19**.**

Let $M$ and $N$ be structures and consider $Aut(M)$ (resp. $Aut(N)$) as acting naturally on $M$ (resp. $N$). We say that $(Aut(M),M)$ and $(Aut(N),N)$ are isomorphic as permutation groups if there exists a bijection $f:M\rightarrow N$ such that the map $h\mapsto fhf^{-1}$ is an isomorphism from $Aut(M)$ onto $Aut(N)$.

We recall the statement of Theorem 1 and prove it.

Theorem 1**.**

Let $\mathbf{K}_{*}$ be the class of countable structures $M$ satisfying:

1. (1)

$M$* has the strong small index property;* 2. (2)

for every finite $A\subseteq M$, $acl_{M}(A)$ is finite; 3. (3)

for every $a\in M$, $acl_{M}(\{a\})=\{a\}$.

Then for $M,N\in\mathbf{K}_{*}$, $Aut(M)$ and $Aut(N)$ are isomorphic as abstract groups if and only if $(Aut(M),M)$ and $(Aut(N),N)$ are isomorphic as permutation groups.

Proof of Theorem 1.

Let $M,N\in\mathbf{K}_{*}$, and suppose that $F:Aut(M)\cong Aut(N)$. Passing to canonical relational structures (cf. [2, pg. 26]), i.e. adding a relation symbol $R$ of arity $n$ for every $n$-ary $Aut(M)$-orbit $\Omega\subseteq M^{n}$, we can assume without loss of generality that $M$ and $N$ are homogeneous. Now, by Theorem 16(2), we have that the isomorphism $F$ induces the isomorphism:

[TABLE]

In particular, $\hat{F}$ maps $P^{min}_{\mathbf{A}(M)}$ onto $P^{min}_{\mathbf{A}(N)}$. Furthermore, by Proposition 18 we have:

[TABLE]

Hence, $\hat{F}$ induces the bijection $f:M\rightarrow N$ such that (recall that $\hat{F}=\mathbf{j}_{N}^{-1}F\,\mathbf{j}_{M}$):

[TABLE]

Let $G:h\mapsto fhf^{-1}$, for $h\in Aut(M)$. We claim that $G=F$. Let in fact $h\in Aut(M)$, $a,b\in M$ and suppose that $F(h)(f(a))=f(b)$. Then:

[TABLE]

So, $fhf^{-1}(f(a))=fh(a)=f(b)$, as wanted. Hence, $f:M\rightarrow N$ witnesses that $(Aut(M),M)$ and $(Aut(N),N)$ are isomorphic as permutation groups. Notice that the “moreover part” of the theorem is clear from the proof (since $G=F$). ∎

Definition 20**.**

We say that two structures $M$ and $N$ are bi-definable if there is a bijection $f:M\rightarrow N$ such that for every $A\subseteq M^{n}$, $A$ is $\emptyset$-definable in $M$ if and only if $f(A)$ is $\emptyset$-definable in $N$.

Fact 21** ([9], Proposition 1.3).**

Let $M$ and $N$ be countable $\aleph_{0}$-categorical structures. Then the following are equivalent:

1. (1)

$(Aut(M),M)\cong(Aut(N),N)$; 2. (2)

$M$* and $N$ are bi-definable.*

Proof of Corollary 2.

Let $M,N\in\mathbf{K}_{*}$, and suppose that $Aut(M)\cong Aut(N)$. As before, passing to canonical relational structures, we can assume without loss of generality that $M$ and $N$ are homogeneous. Furthermore, since $M$ and $N$ are $\aleph_{0}$-categorical, this passage preserves definability. Now, since $M$ and $N$ are $\aleph_{0}$-categorical and with no algebraicity, the conditions of Theorem 1 are met (cf. Remark 7), and so we have that $(Aut(M),M)\cong(Aut(N),N)$ are isomorphic as permutation groups. Hence, by Fact 21, we are done. Notice that the “moreover part” of the corollary is taken care of by the “moreover part” of Theorem 1. ∎

We now pass to the proof of Theorem 3.

Fact 22** (Frucht’s Theorem [3]).**

Every finite group is the group of automorphisms of a finite graph.

Proof of Theorem 3.

Let $\Gamma$ be a finite graph on vertex set $\{0,...,n-1\}$ and

[TABLE]

be such that the $P_{\ell}$ are unary predicates and the $R_{\ell,k}$ are binary relations. Let $\mathbf{K}_{\Gamma}$ be the class of finite $L_{\Gamma}$-models $M$ such that:

1. (1)

$(P^{M}_{\ell}:\ell<n)$ is a partition of $M$; 2. (2)

$R^{M}_{\ell,k}$ is a symmetric irreflexive relation on $P_{\ell}\times P_{k}$.

Notice that $\mathbf{K}_{\Gamma}$ is a free amalgamation class (cf. [8, Definition 4]). Let $M_{\Gamma}$ be the corresponding countable homogeneous structure. By [8, Corollary 2], $M_{\Gamma}$ has the strong small index property, and, obviously, $M_{\Gamma}$ is $\aleph_{0}$-categorical and has no algebraicity. Using Corollary 2 it is now easy to see that:

[TABLE]

Thus, by Fact 22 we are done. ∎