New \(2\)-closed groups that are not automorphism groups of digraphs (Q6662818)

Let \(G\) be a transitive permutation group on a finite set \(\Omega\). An \textit{orbital digraph for \(G\)} is a pair \((\Omega,E)\), where \(E\) is a \(G\)-orbit of a pair of vertices. Any digraph on which \(G\) acts is a union of orbital digraphs for \(G\). The \textit{\(2\)-closure of \(G\)} is the maximal permutation group \(H\) such that \(G \le H \le \mathrm{Sym}(\Omega)\) and \(H\) defining the same set of orbital digraphs as \(G\). A group is \textit{\(2\)-closed} if it coincides with its \(2\)-closure.\N\NAutomorphism groups of graphs and digraphs are always \(2\)-closed. However, not every \(2\)-closed permutation group is the automorphism group of a digraph. In this paper, the authors give new examples of such \(2\)-closed groups. The interest in these groups is twofold.\N\NFirst, the question addressed can be viewed as a generalization of the investigation into \textit{graphical regular representations} and \textit{digraphical regular representations} conducted by \textit{L. Babai} [Period. Math. Hung. 11, 257--270 (1980; Zbl 0452.05030)] and \textit{C. D. Godsil} [J. Comb. Theory, Ser. B 29, 116--140 (1980; Zbl 0443.05047)], which established which finite abstract group can be the regular automorphism group of some Cayley graph (directed or not).\N\NSecond, this study is related to the \textit{Polycirculant Conjecture}, which has been formulated in two forms. The original version, proposed by \textit{D. Marusic} [Discrete Math. 36, 69--81 (1981; Zbl 0459.05041)], asks whether the automorphism group of every vertex-transitive digraph contains a semiregular element. Independently, \textit{P. Cameron} (ed.) [Discrete Math. 167--168, 605--615 (1997; Zbl 0870.05075)] posed a similar question: does every \(2\)-closed group contain a semiregular permutation? Understanding which \(2\)-closed groups are not automorphism groups of any digraph sheds light on the differences between these two versions of the Polycirculant Conjecture.\N\NLet us introduce the groups which are the main focus of this paper.\N\NIn [J. Comb. Theory, Ser. B 158, Part 2, 176--205 (2023; Zbl 1522.20013)], \textit{M. Giudici} et al. classified the \(2\)-closed primitive permutation groups of rank \(4\) that are neither contained in \(\mathrm{A}\Gamma\mathrm{L}_1(q)\) nor the automorphism group of any digraph. These groups fall into two infinite families, along with seven sporadic examples. The first family consists of the groups defined by\N\[\N\mathbf{G}(m,3) := V \rtimes D_8 \circ \mathrm{GL}_m(3) ) \le \mathrm{AGL}_{2m}(3) \,,\N\]\Nwhere\N\begin{itemize}\N\item \(m \ge 2\) is an integer,\N\item \(V\) is the regular subgroup of the affine group \(\mathrm{AGL}_{2m}(3)\),\N\item \(D_8 \circ \mathrm{GL}_m(3)\) preserves the tensor product decomposition \(V = X \otimes Y\),\N\item \(X\) is a \(2\)-dimensional vector space over \(\mathbb{F}_3\) on which \(D_8\) preserves a direct sum decomposition,\N\item and \(Y\) is an \(m\)-dimensional vector space over \(\mathbb{F}_3\) on which \(\mathrm{GL}_m(3)\) acts naturally.\N\end{itemize}\NBy varying the characteristic in this definition, one can build \(\mathbf{G}(m,p)\), for every prime \(p\).\N\NThe main result of the paper is that \(\mathbf{G}(m,p)\) is a \(2\)-closed group that is not the automorphism group of any digraph if and only if \(p\in \{3,5,7,13\}\). Moreover, for these primes, the permutational ranks of \(\mathbf{G}(m,p)\) are \(4\), \(5\), \(5\) and \(7\), respectively. It remains open whether examples exist of arbitrarily large rank.