Odd order flag-transitive affine planes of dimension three over their kernel

Abstract

With the exception of Hering's plane of order 27, all known odd order flag-transitive a‰ne planes are one of two types: admitting a cyclic transitive action on the line at infinity, or admitting a transitive action on the line at infinity with two equal-sized cyclic orbits. In this paper we show that when the dimension over the kernel for these planes is three, then the known examples are the only possibilities for either of these two types. Moreover, subject to a relatively mild gcd condition, one of these two actions must occur. Hence, subject to this gcd condition, all odd order three-dimensional flag-transitive a‰ne planes have been classified.