Line-transitive point-imprimitive linear spaces: the grid case

Abstract

For a fixed integer $k$, all but a finite number of line-transitive linear spaces with lines of size $k$ are point-primitive. In this paper, we study the finite class of examples where a line-transitive group is point-imprimitive, that is, preserves a non-trivial partition of the point set. We restrict to the case where (i) the number of unordered point-pairs, on a given line, contained in the same class of the partition is at most eight, and (ii) some non-identity group element fixes setwise each class of the partition, and also fixes a point. This family of linear spaces was studied by Ngo Dac Tuan and the third author in 2003, leaving several problems unresolved. We prove that all examples in this family are known, namely Desarguesian projective planes of appropriate orders, and an additional example on $91$ points. The result is obtained by a combination of theoretical analysis, and exhaustive computer search.