A classification of finite partial linear spaces with a primitive rank 3 automorphism group of almost simple type

Abstract

A partial linear space is a non-empty set of points, provided with a collection of subsets called lines such that any pair of points is contained in at most one line and every line contains at least two points. Graphs and linear spaces are particular cases of partial linear spaces. A partial linear space which is not a graph or a linear space is called proper. In this paper, we give a complete classification of all finite proper partial linear spaces admitting a primitive rank 3 automorphism group of almost simple type.

Unitals Admitting All Translations

“…Below we require only the knowledge of the (sub)degrees, as listed in [6]. These permutation groups have been classified by Bannai, Kantor, and Liebler, and Liebeck and Saxl; the completeness of the list depends on the classification of finite simple groups.…”

“…Then k and are both divisible by some prime power Q > 1, and k + l > Q 3 , compare the list in [6]. Then k and are both divisible by some prime power Q > 1, and k + l > Q 3 , compare the list in [6].…”

“…Thus, the order of the center Z ofG L divides gcd(3, f + 1) = Z (SU(3, f )). (4) G L ∼ = Sz(2 s ) with q = 2 2s ≥ 2 6 . Let ∼ = SL(2, f ) denote the corresponding subgroup ofĜ L .…”

Extremely primitive groups and linear spaces

“…A natural generalization is the classification of linear spaces admitting a primitive rank 3 automorphism group. Devillers classified these linear spaces when the automorphism groups are of almost simple type and grid type [10], [9]. In recent works [1], [16], Biliotti, Francot and Montinaro have completed the classification in the case when the automorphism groups are of affine type.…”

Rank 3 Latin square designs

“…Significant progress has been made when the group is assumed to be primitive [5]. Here we consider the most basic imprimitive case.…”