Anton BETTEN, Anne DELANDTSHEER
The paper summarises existing theory and classifications for finite line-transitive linearspaces,develops the theory further,and organises it in a way that enables its effective application.Thestarting point is a theorem of Camina and the fifth author that identifies three kinds of line-transitiveautomorphism groups of linear spaces.In two of these cases the group may be imprimitive on points,that is,the group leaves invariant a nontrivial partition of the point set.In the first of these casesthe group is almost simple with point-transitive simple socle,and may or may not be point-primitive,while in the second case the group has a non-trivial point-intransitive normal subgroup and hence isdefinitely point-imprimitive.The theory presented here focuses on point-imprimitive groups.As anon-trivial application a classification is given of the point-imprimitive,line-transitive groups,and thecorresponding linear spaces,for which the greatest common divisor gcd(k,v-1)≤8,where v is thenumber of points,and k is the line size.Motivation for this classification comes from a result of WeidongFang and Huiling Li in 1993,that there are only finitely many non-trivial point-imprimitive,line-transitive linear spaces for a given value of gcd(k,v-1).The classification strengthens the classificationby Camina and Mischke under the much stronger restriction k≤8:no additional examples arise.The paper provides the backbone for future computer-based classifications of point-imprimitive,line-transitive linear spaces with small parameters.Several suggestions for further investigations are made.