Inequalities involving the Delandtsheer--Doyen parameters for finite line-transitive linear spaces
Let \(\mathcal{L}\) be a linear space with a line-transitive, point-imprimitive automorphism group \(G\). In [\textit{A. Delandtsheer} and \textit{J. Doyen}, Geom. Dedicata 29, 307-310 (1989; Zbl 0673.05010)] two integer parameters \(x\) and \(y\) were introduced for every \(G\)-invariant nontrivial partition of the point set. The paper explores inequalities involving these two parameters. There is special interest to the case where the \(G\)-invariant partition \(\mathcal{C}\) is \(G\)-normal, that is, to the case where \(\mathcal{C}\) is the set of orbits of a normal subgroup \(K\) of \(G\). If the constant line size is greater than \(2x+\frac{3}{2}+\sqrt{4x-\frac{7}{4}}\), then the authors show that \(K\) acts semiregularly on points and on lines. If the parameter \(x\) is at most 8, then it is shown that either \(K\) is abelian and semiregular on points or \(\mathcal{L}\) is one of four explicitly known examples.
scientific article; zbMATH DE number 1917907