On flag-transitive c.c*-geometries

Baumeister, Barbara

doi:10.1515/9783110893106.3

Cited by 13 publications

(33 citation statements)

References 9 publications

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“…(Note that H · Alt(6) also acts flag-transively on ( p, O).) It follows from [4] (Theorem B, (3) (ii)) that ( p, O) is simply connected.…”

Section: The Two Jvt-geometriesmentioning

confidence: 98%

“…Then admits just one 1-factorization χ, which can be constructed as follows ( [7,17]). We can assume that V = H , for a hyperoval H of PG (2,4). As set of colours we take a line L of PG (2,4) external to H and, given any two distinct points a, b ∈ H , we define χ({a, b}) as the meet point of L with the line of PG(2, 4) joining a with b.…”

Section: On 1-factorizations Of Complete Graphsmentioning

confidence: 99%

“…Let C be the set of line of PG (3,4) joining p with points of O and let C = L∈C L. We take P = C\({ p} ∪ O) as the set of points of ( p, O). As planes we take the planes u of PG (3,4) such that p ∈ u and u ∩ O = ∅.…”

Section: The Two Jvt-geometriesmentioning

confidence: 99%

“…As planes we take the planes u of PG (3,4) such that p ∈ u and u ∩ O = ∅. Two points of P not on the same line of C are said to form a line of ( p, O).…”

Section: The Two Jvt-geometriesmentioning

confidence: 99%

“…Two points of P not on the same line of C are said to form a line of ( p, O). The incidence relation is the natural one, inherited from PG (3,4). It is straightforward to check that ( p, O) is a flag-transitive c.c * -geometry of order 4.…”

Section: The Two Jvt-geometriesmentioning

confidence: 99%

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Baumeister

Pasini

1996

Journal of Algebraic Combinatorics

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Abstract.We study flat flag-transitive c.c * -geometries. We prove that, apart from one exception related to Sym(6), all these geometries are gluings in the meaning of [6]. They are obtained by gluing two copies of an affine space over GF(2). There are several ways of gluing two copies of the n-dimensional affine space over GF(2). In one way, which deserves to be called the canonical one, we get a geometry with automorphism group G = 2 2n · L n (2) and covered by the truncated Coxeter complex of type D 2 n . The non-canonical ways give us geometries with smaller automorphism group (G ≤ 2 2n · (2 n − 1)n) and which seldom (never ?) can be obtained as quotients of truncated Coxeter complexes.

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“…(Note that H · Alt(6) also acts flag-transively on ( p, O).) It follows from [4] (Theorem B, (3) (ii)) that ( p, O) is simply connected.…”

Section: The Two Jvt-geometriesmentioning

confidence: 98%

Section: On 1-factorizations Of Complete Graphsmentioning

confidence: 99%

Section: The Two Jvt-geometriesmentioning

confidence: 99%

“…As planes we take the planes u of PG (3,4) such that p ∈ u and u ∩ O = ∅. Two points of P not on the same line of C are said to form a line of ( p, O).…”

Section: The Two Jvt-geometriesmentioning

confidence: 99%

Section: The Two Jvt-geometriesmentioning

confidence: 99%

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Baumeister

Pasini

1996

Journal of Algebraic Combinatorics

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The Universal Covers of the Sporadic Semibiplanes

Baumeister

Ṗasechnik

1996

European Journal of Combinatorics

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We determine the universal covers of the few flag-transitive sporadic semibiplanes . They were already known by computer-aided coset enumeration . The method we are using seems to be new and of interest on its own . ÷ 1996 Academic Press Limited 1 . I NTRODUCTION This paper is a continuation of [2 , 3] . In [2 , 3] all known examples of flag-transitive c и c *-geometries , also called semibiplanes , were listed and all such geometries satisfying certain extra assumption were classified . The universal covers of certain c и c *-geometries were found using computer-aided coset enumeration . One aim of this paper is to determine the universal covers of these geometries without using a computer . Another aim is to exhibit some of the technique used , which seems to be new and of interest on its own . Roughly speaking , if (a) there is a good theoretic bound on the number of elements of the (flag-transitive) geometry , and (b) for any flag-transitive cover Ᏻ ˜ of the geometry Ᏻ , there exists a perfect flag-transitive group of automorphisms G ˜ of Ᏻ ˜ , then there is a good chance that the automorphism group of the universal cover contains a flag-transitive subgroup G ˜ , which is a perfect central extension of G . It turns out that , for the c и c *-geometries considered here , G ˜ is such an extension and , moreover , G ˜ / Z ( G ˜ ) is a simple group . Since the Schur multipliers of the finite simple groups are known , we are able to determine G ˜ and thereby the universal cover .In the Appendix we give the distribution diagrams of the point -circle incidence graphs of the sporadic c и c *-geometries , although only a small part of the information they carry is actually used here .Our terminology is fairly standard ; see the next section . The elementsof Ᏻ are called points , lines and circles .

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Flag-transitive Extensions of Dual Affine Planes

Pasini

1996

European Journal of Combinatorics

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On flag-transitive c.c*-geometries

Cited by 13 publications

References 9 publications

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The Universal Covers of the Sporadic Semibiplanes

Flag-transitive Extensions of Dual Affine Planes

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