A maximal subgroup of \(PSp_ 6(2^ m)\) related to a spread
type: Item
instance of: scholarly article
title: A maximal subgroup of \(PSp_ 6(2^ m)\) related to a spread
zbMATH Open document ID: 0574.20011
DOI: 10.1016/0021-8693(83)90071-6
published in: Journal of Algebra
publication date: 1983-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 20D06
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 51A40
Mathematics Subject Classification ID: 14L35
zbMATH DE Number: 3916543
zbMATH Keywords: projective space
zbMATH Keywords: spread
zbMATH Keywords: totally isotropic planes
zbMATH Keywords: symplectic polarity
zbMATH Keywords: maximal subgroup
zbMATH Keywords: orthogonal polarity
author: Roger H. Dye
MaRDI profile type: Publication
full work available at URL: 0021-8693(83)90071-6
OpenAlex ID: W2072958884
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description: scientific article; zbMATH DE number 3916543