Finite Line-transitive Linear Spaces: Theory and Search Strategies

The paper summarises existing theory and classifications for finite line-transitive linear spaces, develops the theory further, and organises it in a way that enables its effective application. The starting point is a theorem of Camina and the fifth author that identifies three kinds of line-transitive automorphism groups of linear spaces. In two of these cases the group may be imprimitive on points, that is, the group leaves invariant a nontrivial partition of the point set. In the first of these cases the group is almost simple with point-transitive simple socle, and may or may not be point-primitive, while in the second case the group has a non-trivial point-intransitive normal subgroup and hence is definitely point-imprimitive. The theory presented here focuses on point-imprimitive groups. As a non-trivial application a classification is given of the point-imprimitive, line-transitive groups, and the corresponding linear spaces, for which the greatest common divisor gcd(k, v − 1) ≤ 8, where v is the number of points, and k is the line size. Motivation for this classification comes from a result of Weidong Fang and Huiling Li in 1993, that there are only finitely many non-trivial point-imprimitive, line-transitive linear spaces for a given value of gcd(k, v −1). The classification strengthens the classification by Camina and Mischke under the much stronger restriction k ≤ 8: no additional examples arise. The paper provides the backbone for future computer-based classifications of point-imprimitive, line-transitive linear spaces with small parameters. Several suggestions for further investigations are made.