The elation group of a 4-dimensional Laguerre plane

1991 ◽  
Vol 111 (3) ◽  
pp. 207-231 ◽  
Author(s):  
G�nter F. Steinke
Keyword(s):  
Laguerre Plane ◽  
Elation Group

scholarly journals A FLAT LAGUERRE PLANE OF KLEINEWILLINGHÖFER TYPE V

2011 ◽  
Vol 91 (2) ◽  
pp. 257-274 ◽  
Author(s):  
JEROEN SCHILLEWAERT ◽  
GÜNTER F. STEINKE
Keyword(s):  
Parameter Family ◽  
Laguerre Plane ◽  
The Other ◽  
Flat Laguerre Plane ◽  
Laguerre Planes ◽  
Type V ◽  
Central Automorphisms ◽  
Kleinewillinghöfer Type

AbstractThe Kleinewillinghöfer types of Laguerre planes reflect the transitivity properties of certain groups of central automorphisms. Polster and Steinke have shown that some of the conceivable types for flat Laguerre planes cannot exist and given models for most of the other types. The existence of only a few types is still in doubt. One of these is type V.A.1, whose existence we prove here. In order to construct our model, we make systematic use of the restrictions imposed by the group. We conjecture that our example belongs to a one-parameter family of planes all of type V.A.1.

A Characterization of Some Geometries of Chains

1974 ◽  
Vol 26 (02) ◽  
pp. 257-272 ◽  
Author(s):  
Yi Chen
Keyword(s):  
Projective Plane ◽  
Minkowski Plane ◽  
Close Relation ◽  
Laguerre Plane ◽  
Geometric Structures ◽  
Möbius Plane

The geometries considered here are the Möbius plane M() (W. Benz [1]), the Laguerre plane L() (W. Benz and H. Mäurer [7]) and the Minkowski plane A() (W. Benz [5], G. Kaerlein [18]) over a field . All of them are geometries of an algebra with identity over a field. The characterization of the projective plane over a field by the proposition of Pappus first gave a close relation between algebraic and geometric structures. B. L. v. d. Waedern and L. J. Smid [28] presented a further example by characterizing the Möbius and Laguerre plane with incidence axioms and the "complete" proposition of Miquel.

scholarly journals The inner and outer space of 2-dimensional Laguerre planes

1997 ◽  
Vol 62 (1) ◽  
pp. 104-127 ◽  
Author(s):  
B. Polster ◽  
G. F. Steinke
Keyword(s):  
Outer Space ◽  
Laguerre Plane ◽  
Generalized Quadrangles ◽  
Base Points ◽  
Laguerre Planes

AbstractThe classical 2-dimensional Laguerre plane is obtained as the geometry of non-trivial plane sections of a cylinder in R3 with a circle in R2 as base. Points and lines in R3 define subsets of the circle set of this geometry via the affine non-vertical planes that contain them. Furthemore, vertical lines and planes define partitions of the circle set via the points and affine non-vertical lines, respectively, contained in them.We investigate abstract counterparts of such sets of circles and partitions in arbitrary 2-dimensional Laguerre planes. We also prove a number of related results for generalized quadrangles associated with 2-dimensional Laguerre planes.

scholarly journals Flat Laguerre planes of Kleinewillinghöfer type E obtained by cut and paste

2005 ◽  
Vol 72 (2) ◽  
pp. 213-223 ◽  
Author(s):  
Günter F. Steinke
Keyword(s):  
Laguerre Plane ◽  
Flat Laguerre Plane ◽  
Laguerre Planes ◽  
Kleinewillinghöfer Type

We provide examples of flat Laguerre planes of Kleinewillinghöfer type E, thus completing the classification of flat Laguerre planes with respect to Laguerre translations in B. Polster and G.F. Steinke, Results Maths. (2004). These planes are obtained by a method for constructing a new flat Laguerre plane from three given Laguerre planes devised in B. Polster and G. Steinke, Canad. Math. Bull. (1995) but no examples were given there.

scholarly journals The Double Tangency Symmetries in Laguerre Plane

2007 ◽  
Vol 55 (2) ◽  
pp. 123-137
Author(s):  
Jarosław Kosiorek ◽  
Andrzej Matraś
Keyword(s):  
Laguerre Plane

scholarly journals On ${\rm STD}_6[18,3]$'s and ${\rm STD}_7[21,3]$'s Admitting a Semiregular Automorphism Group of Order 9

10.37236/237 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Kenzi Akiyama ◽  
Masayuki Ogawa ◽  
Chihiro Suetake
Keyword(s):  
Automorphism Group ◽  
Group Ring ◽  
Elation Group ◽  
Transversal Designs ◽  
Semiregular Automorphism

We characterize symmetric transversal designs ${\rm STD}_{\lambda}[k,u]$'s which have a semiregular automorphism group $G$ on both points and blocks containing an elation group of order $u$ using the group ring ${\bf Z}[G]$. Let $n_\lambda$ be the number of nonisomorphic ${\rm STD}_{\lambda}[3\lambda,3]$'s. It is known that $n_1=1,\ n_2=1,\ n_3=4, n_4=1$, and $n_5=0$. We classify ${\rm STD}_6[18,3]$'s and ${\rm STD}_7[21,3]$'s which have a semiregular noncyclic automorphism group of order 9 on both points and blocks containing an elation of order 3 using this characterization. The former case yields exactly twenty nonisomorphic ${\rm STD}_6[18,3]$'s and the latter case yields exactly three nonisomorphic ${\rm STD}_7[21,3]$'s. These yield $n_6\geq20$ and $n_7\geq 5$, because B. Brock and A. Murray constructed two other ${\rm STD}_7[21,3]$'s in 1991. We used a computer for our research.

A non-miquelian Laguerre plane satisfying a theorem of miquelian type

1998 ◽  
Vol 61 (1-2) ◽  
pp. 32-38 ◽  
Author(s):  
Olaf Br�cker
Keyword(s):  
Laguerre Plane
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