Maximal subgroups of \(PSp_{6n}(q)\) stabilizing spreads of totally isotropic planes
In [Ann. Mat. Pura Apl., IV. Ser. 114, 173-192 (1977; Zbl 0369.50012)] the author constructed a spread \({\mathcal S}_ r\) of PG(2r-1,q) consisting of totally isotropic PG(r-1,q) of a symplectic polarity and proved that the stabilizer of \({\mathcal S}_ r\) in the group \(PSp_{2nr}(q)\) of the polarity is \(PSp_{2n}(q^ r)\cdot <\sigma >\), where \(\sigma\) is the collineation of order r of the \(PG(2n-1,q^ r)\) on which \(PSp_{2n}(q^ r)\) acts induced by the automorphism \(\lambda \to \lambda^ q\) of \(GF(q^ r)\). Recall that a spread of PG(m-1,q) is a partition of its points by a set of subspaces all of the same dimension. In [J. Algebra 87, 493-509 (1984)] he proved that if r is composite and t is a prime divisor of r then \(PSp_{2n}(q^ r)\cdot <\sigma >\) is strictly contained in the stabilizer of \({\mathcal S}_ t\), and conjectured that if r is prime then \(PSp_{2n}(q^ r)\cdot <\sigma >\) is maximal in \(PSp_{2nr}(q)\). The conjecture was proved in [loc.cit.] for the case \(r=2\) and in [J. Algebra 84, 128-135 (1983; Zbl 0574.20011)] for the case \(n=1\), \(r=3\) and q even. In the present paper the author proves the conjecture for the case \(r=3\). The proof is geometric and makes crucial use of the fact that \(PSp_ 2(q^ 3)\cdot <\sigma >\) acts transitively on the totally isotropic planes of PG(5,q) that are not in \({\mathcal S}_ 3\). For a prime r greater than 3 there is no analogous fact; the orbits of totally isotropic PG(r-1,q) of PG(2r-1,q) under \(PSp_ 2(q^ r)\cdot <\sigma >\) are then far more complicated. For these reasons the present approach does not extend. The conjecture remains open for primes r greater then 3.
scientific article; zbMATH DE number 3948601