With the exception of Bering's plane of order 27, all known odd order flag-transitive affine 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...

A linear space is an incidence structure of points and lines such that any two points are incident with exactly one line, any point being incident with at least two lines and any line with at least two points. Within a linear space S, we shall often identify a line L with the set of points incident with L and define the size of L to be the number of points of S incident with L. The degree of a ...

Throughout this paper V always will denote a design with "t; points, k > 2 points per line, and>' = 1 line through any two different points. Let G <:: Aut (V). I will primarily be interested in the case in which G either is 2-transitive on the points of VOl' is transitive on the flags (incident point-line pairs) ofV. Note that 2-transitivity implies flag-transitivity since>. = 1. The subject ma...

The finite flag-transitive linear spaces which have an insoluble automorphism group were given a precise description in [BDD+90], and their classification has recently been completed (see [Lie98] and [Sax02]). However, the remaining case where the automorphism group is a subgroup of one-dimensional affine transformations has not been classified and bears a variety of known examples. Here we giv...

This paper deals with flag-transitive non-symmetric 2-designs with (r, λ) = 1. We prove that if D is a non-trivial non-symmetric 2-(v, k, λ) design with (r, λ) = 1 and G 6 Aut(D) is flag-transitive with Soc(G) = An for n > 5, then D is a 2-(6, 3, 2) design, the projective space PG(3, 2), or a 2-(10, 6, 5) design.