Vorlesungen über Geometrie der Algebren

Benz, Walter

doi:10.1007/978-3-642-88670-6

Cited by 273 publications

(143 citation statements)

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“…If it is clear what linear representation we are using, we write simply K n wrḠ. Changing the set of representatives the action on the set of blocks remains the same, but the action induced on a block changes according to (1) 2.2 Now we want discuss the converse. So let G = (G, Ω) be a transitive permutation group endowed with a finite translation system of imprimitivitȳ Ω = {∆ 1 , .…”

Section: 11mentioning

confidence: 99%

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Imprimitive groups highly transitive on blocks

Bartolone

Musumeci²,

Strambach³

2004

Journal of Group Theory

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We classify imprimitive groups acting highly transitively on blocks and satisfying conditions common in geometry. They can be realized as suitable subgroups of twisted wreath products.In contrast to the theory of primitive permutation groups, the literature for imprimitive groups appears needy. The best known construction principle for imprimitive groups is given by wreath products of two groups U andḠ.Suppose that U is a vector space and let V be the direct sum t i=1 U i of t copies of U . IfḠ is a subgroup of the symmetric group S t acting transitively on the set {U 1 , . . . , U t } of components of V and α :Ḡ U t → GL(U ) is a linear representation of the stabilizer inḠ of the component U t , then the twisted wreath product G = U wr αḠ may be seen as a group of affine mappings acting transitively on a suitable system of affine subspaces of V and having t blocks there. The abelian group T consisting of all translations is the corresponding inertial group, i.e. the normal subgroup of G leaving each block fixed. Restricting to aḠ-invariant subspace W of V , for which there is an integer m ≤ t such that the projection W −→ i m r=i 1 U r is an isomorphism with respect to any m-subset of {1, . . . , t}, we obtain a subgroup of G having the same system of blocks as G and inertial group given by the translations corresponding to W . In particular, the inertial group induces a regular action on each block and acts sharply transitively on m-tuples of independent points, i.e. points no two of which lying in the same block. If the representation α is transitive, then U is finite and the stabilizer of a block is 2-transitive on it. * Research supported by M.U.R.S.T..

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Section: 11mentioning

confidence: 99%

“…For instance, the group of automorphisms of a chain geometry ( [1]) is an imprimitive group operating sharply transitively on triples of independent points, such that the stabilizer of a block is 2-transitive on it. This motivated us to study imprimitive permutation groups which are highly transitive on blocks and satisfy conditions common in geometry.…”

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confidence: 99%

Imprimitive groups highly transitive on blocks

Bartolone

Musumeci²,

Strambach³

2004

Journal of Group Theory

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“…By the Benz planes we mean Möbius, Laquerre and Minkowski planes which were jointly discussed in [3].…”

Section: Introductionmentioning

confidence: 99%

“…The systems β 0 and β p (for p = 1, 2) satisfying some conditions are the Möbius, Laguerre and Minkowski planes, 168 H. Makowiecka respectively. We shall use the axiomatic description of the Benz plane ( [15], p. 639-643), which takes into account earlier books and papers ( [3], [5], [18]). …”

Section: Introductionmentioning

confidence: 99%

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On some operations on finite affine planes applied to the theory of regular configurations

Makowiecka¹

2010

Demonstratio Mathematica

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Abstract. We consider some operations on affine planes which resemble the construction of a derived affine plane at a point of the Benz plane. We call them Benz-contractions (B-contractions), distinguishing between chain contractions and generator contractions. We prove that the Pappos-Pascal configuration is the B-contraction of the affine plane of order 4 and we relate it to the Havliček-Tietze configuration. We present a new (HT) 0 -configuration and research some problems of embeddability for (P-P), (H-T), and (HT) 0 . We propose a method of finding (n − 2) regular configurations on an arbitrary affine plane of order n. Among them are pairs of configurations with dual type and each such a pair can be completed with one point and n + 1 lines to the initial plane. We prove that for an arbitrary n odd, the non-existence of the symmetric configurationimplies the non-existence of the projective plane of order n. On the basis of Gropp's article [10], we solve some current problems concerning the existence of non-symmetric configurations with a natural index.

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Affine Ebenen Mit Orthogonalitätsrelation

Struve

1984

Mathematical Logic Qtrly

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Vorlesungen über Geometrie der Algebren

Cited by 273 publications

References 0 publications

Imprimitive groups highly transitive on blocks

Imprimitive groups highly transitive on blocks

On some operations on finite affine planes applied to the theory of regular configurations

Affine Ebenen Mit Orthogonalitätsrelation

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