Abstract. Any automorphism of a Benz plane having at least one fixed point induces a collineation on the projective extension of the residual affine plane with reference to this point. When this collineation is a central automorphism, the initial automorphism is called the central automorphism (or central-axial automorphism, cf.([3]). In this paper we present an analytical description of central automorphisms of a miguelian Laguerre planes with the characteristic different from two. This description is applied to find transitive groups of homotheties and translations of types occuring in the classification theorems of R. Kleinewillinghofer ([2]). Some examples over an arbitrary commutative field are constructed, the other over the finite field Z3 and It is interesting that two types of the Kleinewillinghofer classification ([2]) appear only as automorphism subgroups of finite plane of order three or five. This will give a clear characterization of these planes. Throughaut we assume that the characteristic of a plane is not equal to two.
IntroductionIt is well known that any miguelian Laguerre plane of characteristic not equal to two is described for some commutative field F (charF 2) as an incidence structure of the form L = (P, C, where P = {(a, b) e F X F} U {(a)|a € F*} U {(00)},whereElements of P are called points, elements of C -chains and elements of G -generators. We will denote points by capital Latin letters, chains by small Greek letters and generators by small Latin letters. Points of the form where x,y € F are called proper points, the others -improper points. A generator containing the point A we will denote by A and points passing through one generator -parallel or touching points (notation A\\B). Equivalently, a uniform description of miguelian Laguerre planes we obtain using the ring of dual numbers Dp over the field F. In this description P = P (D F € P(Z>f)}, where P(Dp) and P(F) denote the projective line over the ring Dp and the field F respectively, and R -the set of invertible elements of the ring Dp.It is easy to prove ( , where a is an automorphism of the field F,These bijections can be uniquely extended to automorphisms of a Laguerre plane preserving the generator oo in case 1) and 2). An arbitrary automorphism of a miguelian Laguerre plane is the superposition of automorphisms of kind 1), 2), and 3).A central automorphism of a Laguerre plane is an automorphism, which has at least one fixed point and on the projective extension of the residual plane in this fixed point induces a central collineation. The central automorphisms of a Laguerre plane are the following ([3]):1. A translation that is an automorphism having exactly one point wise fixed generator and an invariant pencil of tangent chains or family generators, 2. A homothety that is an automorphism, having exactly two fixed points and an invariant boundle of chains with vertices in these points,3. An affinity that is an automorphism, having exactly one chain pointwise fixed, two generators point wise fixed, or one generator point...
