Maximal subgroups of finite symplectic groups stabilizing spreads of lines. (Q1858288)

The authors describe the embedding \(\iota\colon\text{U}_n(q^2)\to\text{Sp}_{2n}(q)\) and a certain line spread of \(\text{PG}(2n-1,q)\). Let \(G\) be the group \(\iota(\langle U_n(q^2),\tau\sigma\rangle)\), where \(\sigma\) is the Frobenius automorphism \(\Theta\to\Theta^q\) for \(\Theta\) in \(\text{GF}(q^2)\), and \(\tau\) is a suitable mapping in \(\text{GU}_n(q^2)\). The authors show that \(G\) is the stabilizer of a certain line spread and that it is maximal in the symplectic group \(\text{Sp}_{2n}(q)\), except when \(n=2\) and \(q=3\). Similar results were obtained by \textit{P. Kleidman} and \textit{M. Liebeck} [The subgroup structure of the finite classical groups (Lond. Math. Soc. Lect. Note Ser., Vol. 129, Cambridge University Press, Cambridge), (1990; Zbl 0697.20004)]. The proof given here is geometrical. It supplements research done by R. H. Dye in several papers.