Inequalities involving the Delandtsheer–Doyen parameters for finite line-transitive linear spaces

The paper studies line-transitive, point-imprimitive automorphism groups G of finite linear spaces. In particular, it explores inequalities involving two integer parameters x,y introduced by Delandtsheer and Doyen associated with a given G-invariant partition C of the point set. There is special interest in the case where C is G-normal, that is, C is the set of orbits of a normal subgroup of G. For example, if C is G-normal relative to a normal subgroup K and the line size is greater than 2x+32+4x−74, then K is shown to be semiregular on points and on lines. Also, if C is G-normal relative to K and x⩽8, then either K is abelian and semiregular on points or the linear space is one of four explicitly known examples.