A maximal subgroup of \(PSp_ 6(2^ m)\) related to a spread
In [Ann. Mat. Pura Appl., IV. Ser. 114, 173-194 (1977; Zbl 0369.50012)] the author proved that the projective space PG(5,q) has a spread (i.e. a partition of PG(5,q)) \({\mathcal S}\) consisting of \(q^ 3+1\) totally isotropic planes of a symplectic polarity, and that the stabilizer G of \({\mathcal S}\) in the group \(PSp_ 6(q)\) of the polarity is isomorphic to the semidirect product \(PSL_ 2(q^ 3)\cdot <\sigma >\), where \(\sigma\) has order 3 and is induced on the projective line \(PG(1,q^ 3)\) by the field automorphism \(\lambda \to \lambda^ q\) of \(GF(q^ 3)\). Earlier [J. Lond. Math. Soc., II. Ser. 2, 746-748 (1970; Zbl 0206.311)] the author constructed (by different means) \({\mathcal S}\) for the case \(q=2\), and proved that in this case G is maximal in \(PSp_ 6(2)\). Here the author proves that if q is even, then G is a maximal subgroup of \(PSp_ 6(q)\). The proof uses the cataloque of the maximal subgroups of \(PSL_ 4(2^ m)\) obtained by \textit{B. Mwene} [J. Algebra 41, 79-107 (1976; Zbl 0346.20029)] and the fact that an orthogonal polarity in PG(5,q) is symplectic for q even. Therefore, the method of proof can apply only to the case of q even. As a maximal subgroup of \(PSp_ 6(2^ m)\) the subgroup G is of interest because it contains no transvections (elations) and it is a small maximal subgroup of \(PSp_ 6(2^ m)\) except when \(2^ m\) is very small.
scientific article; zbMATH DE number 3916543
scientific article