Line-transitive, point quasiprimitive automorphism groups of finite linear spaces are affine or almost simple

Summary.

The paper reports on an investigation of the structure of line-transitive automorphism groups of a finite linear space. It follows from a result of Richard Block that such a group G is also point-transitive. It has been proved by several people independently that, if in addition G is transitive on flags (incident point-line pairs), then G acts primitively on points and is either an almost simple group or a group of affine transformations of a finite vector space. Recently the first author proved that the conclusion that G is almost simple or affine holds under the weaker assumptions that G is line-transitive and point-primitive. We strengthen this result, proving that G is almost simple or affine under the (even weaker) assumptions that G is line-transitive and point-quasiprimitive. (A permutation group is quasiprimitive if every nontrivial normal subgroup is transitive.) This result is best possible since several examples are known of line-transitive automorphism groups G which are not quasiprimitive on points, and are neither almost simple nor affine.