Ja n 20 07 Linear spaces with a line - transitive point - imprimitive automorphism group and Fang - Li parameter gcd ( k , r ) at most eight ∗

In 1991, Weidong Fang and Huiling Li proved that there are only finitely many non-trivial linear spaces that admit a line-transitive, point imprimitive group action, for a given value of gcd(k, r), where k is the line size and r is the number of lines on a point. The aim of this paper is to make that result effective. We obtain a classification of all linear spaces with this property having gcd(k, r) ≤ 8. To achieve this we collect together existing theory, and prove additional theoretical restrictions of both a combinatorial and group theoretic nature. These are organised into a series of algorithms that, for gcd(k, r) up to a given maximum value, return a list of candidate parameter values and candidate groups. We examine in detail each of the possibilities returned by these algorithms for gcd(k, r) ≤ 8, and complete the classification in this case. 2000 Mathematics Subject Classification: 05B05, 05B25, 20B25