Finite line-transitive linear spaces: parameters and normal point-partitions

Until the 1990’s the only known finite linear spaces admitting line-transitive, pointimprimitive groups of automorphisms were Desarguesian projective planes and two linear spaces with 91 points and line size 6. In 1992 a new family of 467 such spaces was constructed, all having 729 points and line size 8. These were shown to be the only linear spaces attaining an upper bound of Delandtsheer and Doyen on the number of points. Projective planes, and the linear spaces just mentioned on 91 or 729 points, are the only known examples of such spaces, and in all cases the line-transitive group has a non-trivial normal subgroup intransitive on points. The orbits of this normal subgroup form a partition of the point set called a normal point-partition. We give a systematic analysis of finite line-transitive linear spaces with normal point-partitions. As well as the usual parameters of linear spaces there are extra parameters connected with the normal point-partition that a¤ect the structure of the linear space. Using this analysis we characterise the line-transitive linear spaces for which the values of various of these parameters are small. In particular we obtain a classification of all imprimitive linetransitive linear spaces that ‘nearly attain’ the Delandtsheer–Doyen upper bound.