The complete Phan-type theorem for Sp(2n,q)

Gramlich, Ralf; Horn, Max; Nickel, Werner

doi:10.1515/jgt.2006.039

Cited by 10 publications

(14 citation statements)

References 7 publications

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“…We remark that the cases of diagrams A n , C n , and D n have been dealt with previously (see [P1,P2], and also [BS,GHS,G1,H,GHN,GHNS]). So our present result completes the last series of Phan-type results for classical groups, see details below.…”

Section: Main Theorem Bmentioning

confidence: 99%

Odd-dimensional orthogonal groups as amalgams of unitary groups. Part 1: General simple connectedness

et al. 2007

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We extend the Phan theory described in [C. Bennett, R. Gramlich, C. Hoffman, S. Shpectorov, CurtisPhan-Tits theory, in: A.A. Ivanov, M.W. Liebeck, J. Saxl (Eds.), Groups, Combinatorics, and Geometry, World Scientific, River Edge, 2003, pp. 13-29] to the last remaining infinite series of classical Chevalley groups over finite fields. Namely, we prove that the twin buildings for the group Spin(2n + 1, q 2 ), q odd, admit a unique unitary flip and that the corresponding flipflop geometry is simply connected for almost all finite fields F q 2 . Applying standard methods from amalgam theory, this results in a characterization of central quotients of the group Spin(2n + 1, q) by a Phan system of rank one and rank two subgroups. In the present first part of a series of two articles we present simple connectedness results for sufficiently large fields or sufficiently large rank. To be precise, the result stated in the present paper is proved for all cases but n = 3 and q ∈ {3, 5, 7, 9}, the remaining cases are dealt with in the sequel [R. Gramlich, M. Horn, W. Nickel, Odd-dimensional orthogonal groups as amalgams of unitary groups. Part 2: Machine computations, submitted for publication] computationally.

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Section: Main Theorem Bmentioning

confidence: 99%

Odd-dimensional orthogonal groups as amalgams of unitary groups. Part 1: General simple connectedness

et al. 2007

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“…A concrete description of standard pairs of SU 3 (q 2 ) can be found in Section 3.2.1. For a concrete description of standard pairs of Sp 4 (q) see [72,74].…”

Section: Weak Phan Systems Of Arbitrary Spherical Typementioning

confidence: 99%

“…By [72] (with the missing cases dealt with in [74,84]) the geometry G τ is almost always simply connected, resulting in the following Phan-type theorem.…”

Section: Phan-type Theorem Of Type C Nmentioning

confidence: 99%

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Developments in finite Phan theory

Gramlich¹

2009

Innov. Incidence Geom.

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“…In Sect. 3, we analyze the amalgam of rank two parabolics for (n, q) = (3, 3) by applying methods from computer algebra, similar to those previously used by the author in [10,14] (for C n ) and [11] (for B n ; see also [4]). It is assumed throughout the paper that the reader is familiar with the concept of amalgams (refer to [17] for an introduction to the subject).…”

Section: Main Theoremmentioning

confidence: 99%

On the Phan system of the Schur cover of SU(4, 32)

Horn

2008

Des. Codes Cryptogr.

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This article is part of the program described in Bennett et al. (in: Ivanov et al. (ed.) Groups, combinatorics, and geometry, 2003) [3]. We study the Phan amalgams and their universal completions that occur for q = 3 in rank n = 3 for the diagram A 3 = D 3 , corresponding to SU(4, 3 2 ) ∼ = Spin − (6, 3). We show that the associated geometries admit universal 9-fold coverings, by showing that the universal completion of the Phan amalgam is the central extension of SU(4, 3 2 ) by its Schur multiplier. This information provides the last missing piece of information in the full classification of Phan amalgams and their universal completions for A n and D n .

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The complete Phan-type theorem for Sp(2n,q)

Cited by 10 publications

References 7 publications

Odd-dimensional orthogonal groups as amalgams of unitary groups. Part 1: General simple connectedness

Odd-dimensional orthogonal groups as amalgams of unitary groups. Part 1: General simple connectedness

Developments in finite Phan theory

On the Phan system of the Schur cover of SU(4, 32)

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