Book contents
- Frontmatter
- Contents
- PREFACE
- Introduction
- Generalized Steiner systems of type 3-(v, {4,6}, 1)
- Some remarks on D.R. Hughes' construction of M12 and its associated designs
- On k-sets of class [0,1,2,n]2 in PG(r,q)
- Covering graphs and symmetric designs
- Arcs and blocking sets
- Flat embeddings of near 2n-gons
- Codes, caps and linear spaces
- Geometries originating from certain distance-regular graphs
- Transitive automorphism groups of finite quasifields
- On k-sets of type (m,n) in projective planes of square order
- On k-sets of type (m,n) in a Steiner system S(2, l, v)
- Some translation planes of order 81
- A new partial geometry constructed from the Hoffman-Singleton graph
- Locally cotriangular graphs
- Coding theory of designs
- On shears in fixed-point-free affine groups
- On (k,n)-arcs and the falsity of the Lunelli-Sce conjecture
- Cubic surfaces whose points all lie on their 27 lines
- Existence results for translation nets
- Translation planes having PSL(2,w) or SL(3,w) as a collineation group
- Sequenceable groups: a survey
- Polar spaces embedded in a projective space
- On relations among the projective geometry codes
- Partition loops and affine geometries
- Regular cliques in graphs and special 1½ designs
- Bericht über Hecke Algebren und Coxeter Algebren eindlicher Geometrien
- On buildings and locally finite Tits geometries
- Moufang conditions for finite generalized quadrangles
- Embedding geometric lattices in a projective space
- Coverings of certain finite geometries
- On class-regular projective Hjelmslev planes
- On multiplicity-free permutation representations
- On a characterization of the Grassmann manifold representing the lines in a projective space
- Affine subplanes of projective planes
- Point stable designs
- Other talks
- Participants
On class-regular projective Hjelmslev planes
Published online by Cambridge University Press: 05 April 2013
- Frontmatter
- Contents
- PREFACE
- Introduction
- Generalized Steiner systems of type 3-(v, {4,6}, 1)
- Some remarks on D.R. Hughes' construction of M12 and its associated designs
- On k-sets of class [0,1,2,n]2 in PG(r,q)
- Covering graphs and symmetric designs
- Arcs and blocking sets
- Flat embeddings of near 2n-gons
- Codes, caps and linear spaces
- Geometries originating from certain distance-regular graphs
- Transitive automorphism groups of finite quasifields
- On k-sets of type (m,n) in projective planes of square order
- On k-sets of type (m,n) in a Steiner system S(2, l, v)
- Some translation planes of order 81
- A new partial geometry constructed from the Hoffman-Singleton graph
- Locally cotriangular graphs
- Coding theory of designs
- On shears in fixed-point-free affine groups
- On (k,n)-arcs and the falsity of the Lunelli-Sce conjecture
- Cubic surfaces whose points all lie on their 27 lines
- Existence results for translation nets
- Translation planes having PSL(2,w) or SL(3,w) as a collineation group
- Sequenceable groups: a survey
- Polar spaces embedded in a projective space
- On relations among the projective geometry codes
- Partition loops and affine geometries
- Regular cliques in graphs and special 1½ designs
- Bericht über Hecke Algebren und Coxeter Algebren eindlicher Geometrien
- On buildings and locally finite Tits geometries
- Moufang conditions for finite generalized quadrangles
- Embedding geometric lattices in a projective space
- Coverings of certain finite geometries
- On class-regular projective Hjelmslev planes
- On multiplicity-free permutation representations
- On a characterization of the Grassmann manifold representing the lines in a projective space
- Affine subplanes of projective planes
- Point stable designs
- Other talks
- Participants
Summary
Regular projective Hjelslev planes (PH-planes), i.e. PH-planes admitting an automorphism group which is regular on the points and blocks (i.e. a Singer group) were introduced and studied by Jungnickel. In this paper, we study (t,r)-PH-planes admitting an automorphism group which is regular on each point and block class (neighbourhood). We prove that this notion is equivalent to the existence of a projective plane of order r and a (t,r)- Hjelmslev matrix in an abelian group of order t2 as defined by Jungnickel. This enables us to construct many class-regular PHplanes which do not admit a Singer group. Further, all the results of Jungnickel (on regular PH-planes) also hold for class-regular PH-planes.
REMARKS: Projective Hjelmslev planes (H-planes) (more generally, projective Klingenberg planes or K-planes) are generalizations of projective planes in which two points (resp. two lines) are allowed to have more than one line passing through them (resp. more than one point of intersection) and which admit an epimorphism onto a projective plane. For a nice introduction to H-planes (and K-planes) and their automorphism groups, we refer to [3], [4] and [5] ; [1] gives an extensive bibliography of the literature on H-planes and related structures.
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- Finite Geometries and DesignsProceedings of the Second Isle of Thorns Conference 1980, pp. 332 - 336Publisher: Cambridge University PressPrint publication year: 1981