A classification of Laguerre near-planes of order four

The automorphism groups of the Laguerre near-planes of order four

Finite Möbius near-planes

We introduce finite Mobius near-planes of order n and show that these planes uniquely extend to Mobius planes of the same order if n ;:::: 5. Furthermore, Mobius near-planes of order n :s; 4 are discussed and the situation for the other two types of finite circle near-planes, Laguerre and Minkowski near-planes, is reviewed.

Finite Laguerre Near-planes of Odd Order Admitting Desarguesian Derivations

From this definition it readily follows that a Laguerre plane of order n has n + 1 generators, that every circle contains exactly n + 1 points and that there are n3 circles. All known models of finite Laguerre planes are of the following form. Let O be an oval in the Desarguesian projective plane P2 = PG(2, pm), p a prime. Embed P2 into threedimensional projective space P3 = PG(3, pm) and let v...

A characterization of certain elation Laguerre planes

We characterize the non-classical 4-dimensional elation Laguerre planes as precisely those 4-dimensional Laguerre planes of Kleinewillinghöfer type I.D.1. Furthermore, in the class of 2or 4-dimensional Laguerre planes or finite Laguerre planes of odd order, the non-miquelian elation Laguerre planes are precisely the Laguerre planes of Kleinewillinghöfer type D. MSC 2000: 51H15, 51B15, 51E25

Flat Laguerre Planes of Kleinewillinghöfer Type E Obtained by Cut and Paste

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. ...