In this paper we determine all finite Minkowski planes with an automorphism group which satisfies the following transitivity property: any ordered pair of nonparallel points can be mapped onto any other ordered pair of nonparallel points.
1. Introduction
All known finite inversive planes have a two-transitive group of automorphisms. Conversely, every inversive plane admitting an automorphism group which is two-transitive on the points, is of a known type (cf. [9]).
For Minkowski planes the situation is quite similar. All known finite Minkowski planes have an automorphism group acting two-transitively on non-parallel points. In this note we shall show that this property is characteristic for the known Minkowski planes. More precisely, we shall prove the following theorem.
THEOREM. Let Jg be a finite Minkowski plane of odd order n, and suppose that ~ admits an automorphism 9roup F actin9 two-transitively on nonparallel points. Then n is a prime power, Jg ~-Jg (n, (~) for some field automorphism c~ of GF(n), and F contains PSL(2, n) ร PSL(2, n).
For a definition of ~(n, ~b) see Section 2. As Minkowski planes of even order n only exist for n a power of 2 and are unique for given order n = 2 ", this result completes the classification of the Minkowski planes with an automorphism group acting two-transitively on nonparallel points.
2. DEFINITIONS, NOTATION AND BASIC RESULTS
Let M be a set of points and 50 +, 50-, cg three collections of subsets of M. The elements of 50:=50+ w50-are called lines or 9enerators, the elements of cg are called circles. We say that ~ = (M, 50+, 50-, cg) is a Minkowski plane if the following axioms are satisfied (cf. [8] ). (M1): (M2): (M3): (M4): (M5): 50 + and 5 ยฐ-are partitions of M.
]l+nl-[--1 foralll+e50,1e50 -. Given any three points no two on a line, there is a umque passing through these three points. ]lnc[=l for all l~ 50, cECg. There exist three points no two of which are on one line. circle