Extremely primitive classical groups. (Q714111)

A primitive permutation group \(G\) acting on a set \(\Omega\) is said to be `extremely primitive' if a point stabilizer acts primitively on each of its orbits. By a theorem of \textit{A. Mann, C. E. Praeger} and \textit{Á. Seress}, [Groups Geom. Dyn. 1, No. 4, 623-660 (2007; Zbl 1141.20003)], every finite extremely primitive group is either almost simple or of affine type. What is more, in the same paper, Mann et al. give a list of affine-type extremely primitive groups, and they prove that there are only finitely many affine-type extremely primitive groups that do not appear on their list. Attention then turns naturally to the extremely primitive almost simple groups. In the current paper the authors suppose that such a group \(G\) is classical, and they provide a full classification: there are three infinite families of examples (two when \(G\) has socle \(\mathrm{PSL}_2(q)\), one when \(G\) has socle \(\mathrm{PSp}_n(2)'\)) and four individual examples (when \(G\) has socle \(\mathrm{PSL}_4(2)\), \(\mathrm{PSU}_4(3)\), \(\mathrm{PSL}_3(4)\) and \(\mathrm{PSL}_2(11)\), respectively). The body of the proof consists of a case-by-case analysis of the various families given by Aschbacher's theorem on the subgroup structure of classical groups. To perform this analysis, two results are particularly useful: First, a classical theorem of Manning states that if \(G\) is extremely primitive on \(\Omega\) with point-stabilizer \(G_\alpha\), then \(G_\alpha\) is faithful on each of its orbits in \(\Omega\setminus\{\alpha\}\) [\textit{W. A. Manning}, Transactions A. M. S. 29, 815-825 (1927; JFM 53.0108.01)]. The authors use this result to derive information about the center and Fitting subgroup of \(G_\alpha\). Second, the authors make use of the classification of pairs \((G,H)\), where \(G\) is an almost simple classical group, \(H\) is a maximal subgroup of \(G\), and \(H\cap H^x=1\) for some \(x\in G\) [\textit{T. M. Burness, R. M. Guralnick, J. Saxl}, ``Base sizes for finite classical groups'', preprint]. In these cases the action of \(G\) on the cosets \(\Omega=G/H\) cannot be extremely primitive (since a maximal subgroup of \(G\) cannot have prime order), and so such pairs can be excluded. Further families of extremely primitive almost simple groups have been dealt with in subsequent work by the same authors [Bull. Lond. Math. Soc. 44, No. 6, 1147-1154 (2012; Zbl 1264.20001)]. The exceptional groups remain to be treated.