Maximal subgroups of \(PSp_{6n}(q)\) stabilizing spreads of totally isotropic planes
type: Item
instance of: scholarly article
title: Maximal subgroups of \(PSp_{6n}(q)\) stabilizing spreads of totally isotropic planes
zbMATH Open document ID: 0591.20050
DOI: 10.1016/0021-8693(86)90063-3
published in: Journal of Algebra
publication date: 1986-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 20E28
Mathematics Subject Classification ID: 51A40
Mathematics Subject Classification ID: 20D06
Mathematics Subject Classification ID: 14L35
Mathematics Subject Classification ID: 20B25
zbMATH DE Number: 3948601
zbMATH Keywords: spread
zbMATH Keywords: symplectic polarity
zbMATH Keywords: stabilizer
zbMATH Keywords: totally isotropic planes
zbMATH Keywords: orbits
author: Roger H. Dye
MaRDI profile type: Publication
full work available at URL: 0021-8693(86)90063-3
OpenAlex ID: W2004052377
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description: scientific article; zbMATH DE number 3948601