The binary actions of simple groups with a single conjugacy class of involutions (Q6661342)

Let \(G\) be a finite group acting on a set \(\Omega\). The relational complexity of the action is the minimum integer \(k \geq 2\) such that the orbits of \(G\) on \(\Omega^{r}\), for any \(r \geq k\), can be deduced from the orbits of \(G\) on \(\Omega^{k}\). If \(k=2\), then the action of \(G\) on \(\Omega\) is called binary, so, with this measure of complexity, the groups with a binary actions are the simplest permutation groups.\N\NThe primitive binary actions are classified via the O'Nan-Scott Theorem (see [\textit{G. Cherlin}, J. Algebr. Comb. 43, No. 2, 339--374 (2016; Zbl 1378.20001); \textit{J. Wiscons}, Bull. Lond. Math. Soc. 48, No. 2, 291--299 (2016; Zbl 1350.20002)] and the book by the first author et al. [Cherlin's conjecture for finite primitive binary permutation groups. Cham: Springer (2022; Zbl 1515.20011)]). The programme to extend this classification to all transitive binary actions is currently focused on the transitive actions of almost simple groups.\N\NIn the paper under review the authors prove that if \(G\) is a simple group that contains a single conjugacy class of involutions, \(H<G\) is a proper subgroup of \(G\) of even order and \(\Omega\) is the set of right cosets of \(H\) in \(G\), then the action of \(G\) on \(\Omega\) is binary if and only one of the following conditions is fulfilled (a) \(G=\mathsf{SL}_{2}(2^{a})\) and \(H\) is a Sylow 2-subgroup of \(G\); (b) \(G=\mathsf{Sz}(2^{2a+1})\) and \(H\) is the centre of a Sylow 2-subgroup of \(G\); (c) \(G=\mathsf{PSU}_{3}(2^{a}\) and \(H\) is the centre of a Sylow 2-subgroup of \(G\).\N\NThe method of proving the previous result is derived directly from that developed by authors in [``The binary actions of alternating groups'', Preprint, \url{arXiv:2303.06003}]. In particular, in that paper, the authors have introduced a graph \(\Gamma(\mathcal{C})\), where \(\mathcal{C}\) is a conjugacy class of a \(G\): its vertices are elements of \(\mathcal{C}\), and two vertices \(g, h \in \mathcal{C}\) are connected if \([g,h]=1\) and either \(gh^{-1}\) or \(hg^{-1}\) is in \(\mathcal{C}\). They have shown how the study of connected components of \(\Gamma(\mathcal{C})\) sheds considerable light on the possible binary actions for \(G\). In the paper under review, the authors consider the case where \(\mathcal{C}\) is the only class of involutions of \(G\), which greatly simplifies the study of \(\Gamma(\mathcal{C})\).\N\NFurthermore, the authors completely classify all transitive, binary actions of \(\mathsf{PSL}_{2}(q)\) and of \(\mathsf{Sz}(2^{2a+1})\).