Frucht’s theorem without choice

Brian Pinsky

Abstract.

Frucht’s theorem is the statement that “every group is the automorphism group of a graph”. This was shown over ZFC by Sabidussi [sabidussi] and deGroot[degroot], by induction using a well ordered generating set for the group. Sabidussi’s proof is easily modified to use induction on the rank of a generating set, and thus holds over ZF.

We show that Frucht’s theorem is independent of ZFA set theory (ZF with atoms), by showing it fails in several common permutation models. We also present some permutation models where Frucht’s theorem holds, even when AC fails.

As a corollary, we show that a stronger version of Frucht’s theorem due to Babai in [BABAI197826] can fail without choice.

1. Introduction

Frucht’s theorem says every group is isomorphic to the automorphism group of a simple graph (with no loops, multiple edges or directed edges). This was proved by Frucht [frucht] in 1939 for finite groups, and generalized independently by deGroot [degroot] and Sabidussi [sabidussi] to infinite groups in 1959-1960. All the proofs use the same basic idea; given a group $\Gamma$ , take an (edge-)colored directed Cayley graph for $\Gamma$ , with one color for each generator. The automorphism group of this colored Cayley graph is $\Gamma$ , but it is not a simple graph. The constructions in [degroot] and [sabidussi] then add additional structure to encode the colors, as we will summarize in section 2.

Frucht’s theorem for permutation models is more interesting. In section LABEL:counterexamples, we show Frucht’s theorem fails in several permutation models (e.g. the basic Mostowski model). We show this by examining “internal” automorphisms of these models; that is, automorphisms of the atoms which are themselves symmetric. Frucht’s theorem will fail in any model with “enough” internal automorphisms. Theorem LABEL:simplified_no_frucht gives a general criterion for a Frucht’s theorem to fail for a particular group. Since Frucht’s theorem is true in ZF, its failure in these ZFA models can’t be transfered to a symmetric model. This is similar to the results in [jech, chapter 9], but differs in that Frucht’s theorem does not imply AC over ZF.

On the other hand, in Section LABEL:non-counterexamples, we show Frucht’s theorem holds in some permutation models (e.g. the ordered Mostowski model), even when choice fails. The atoms of these models have some structure which becomes rigid in the permutation model, so they have no internal automorphisms. In these cases, we can code the rigid structure of the atoms into rigid graphs, and adapt the proof of Frucht’s theorem from section 2.

In section LABEL:babai, we look at Babai’s theorem [BABAI197826]: if $\Gamma$ is infinite, then $\Gamma$ is the transitive automorphism group of some directed graph (and also an undirected graph with 3 vertex orbits). By applying the Jech-Sochor embedding to a ZFA model where Frucht’s theorem fails, we show Babai’s theorem can fail in ZF.

1.1. Definitions

$V$ is the universe of all sets, and $V_{\alpha}$ is the set of all sets of rank $\alpha$ . In particular, in ZFA, $V_{0}$ is the set of atoms, and $V_{0}=\varnothing$ in ZF.

Inside $V$ , a graph (or “simple graph”) $G$ is a set $\langle V(G),E(G)\rangle$ where $V(G)$ is any set and $E(G)\subseteq V^{2}$ is a symmetric, irreflexive relation. A directed graph only requires $E(G)$ be irreflexive. A colored (directed) graph with a set $S$ of colors is a (directed) graph with an $S$ -valued edge-coloring, i.e. a function from $E(G)$ to $S$ . A pointed graph is a graph with a distinguished vertex.

A homomorphism of graphs is a function on vertices that preserves both edges and non-edges. Homomorphisms must also preserve edge color for colored graphs, and the distinguished vertex for pointed graphs. A rigid graph is one with no non-trivial automorphisms. A graph is connected if any two verticies are connected by a path, i.e. a finite sequence of adjacent edges. A RCPG is a rigid, connected, pointed graph.

Finally, if $S$ is a generating set for a group $\Gamma$ , the corresponding colored directed Cayley graph $\widetilde{\operatorname{Cay}}(\Gamma,S)$ is the colored drected graph with color set $S$ , vertex set $\Gamma$ , and whose $s$ -colored edges are $\{\langle g,gs\rangle|g\in\Gamma\}$ . $\operatorname{Cay}(\Gamma,S)$ is the same graph without the edge colors.

2. Frucht’s theorem in ZF

Theorem 1 (Frucht’s theorem).

Every group is the automorphism group of a graph

It is easy to show every group $\Gamma$ is the automorphism group of the corresponding colored directed Cayley graph $\widetilde{\operatorname{Cay}}(\Gamma,\Gamma)$ . So, to show Frucht’s theorem, we will give two lemmas to replace a colored directed graph with a simple graph.

First, suppose we are given a colored directed graph and several rigid, connected, pointed graphs (RCPGs) to use as labels. By attaching RCPGs to each edge indicating its color and direction, we can produce a simple graph with the same automorphism group, as pictured below (using sticks of length 2 and 3 as labels):

Encoding Directions:
Encoding Colors:

Details of this construction can be found in [details]. This construction requires one rigid connected pointed graph for each color, giving the following:

Lemma 2.

Given an $S$ colored directed graph $G$ and an equinumerous with $S$ set of pairwise non-isomorphic RCPGs, there is a simple graph $G^{\prime}$ with the same automorphism group as $G$

Sabidussi’s construction in [4] works by induction on rank to construct a RCPG for each set. He assumes AC, so he only cares about the graphs for ordinals. However, if we isolate his inductive step, he proves the following:

Lemma 3.

Suppose there is a function $R$ from a set $S$ into pairwise non-isomorphic RCPGs. Then $R$ can be extended to a function from $\mathcal{P}(S)$ into pairwise non-isomorphic RCPGs.

We will sketch the construction, but refer to [details] or [sabidussi] for details. Pick $x\in\mathcal{P}(S)\setminus S$ . We start with 2 complete graphs on $S$ , attached as shown. On the bottom copy, using $R$ , we attach a RCPG to each vertex, making it rigid. On the top copy, we attach degree 1 verticies for each element of $x$ , to encode $x$ . Finally, we attach one distinguished vertex, $t$ , to each top vertex.