Title:

Linetransitive linear spaces

A linear space is an incidence structure consisting of a set of points II and a set of lines L in the power set of P such that any two points are incident with exactly one line. We study those finite linear spaces which admit an automorphism group G which is transitive upon the set of lines of the space. Within the set of all linear spaces lies a particularly important subset: the projective planes. Results exist in the literature [Cam04, CP93] classifying the possible minimal normal subgroups of a group G acting linetransitively on a finite projective plane. We rewrite some of these results to deal with components rather than with minimal normal subgroups. We then prove that, if a group G acts on a projective plane which is not Desarguesian, the G does not contain any components. In order to do this we make use of the classification of finite simple groups; our proof consists of examining the different quasisimple groups given in the classification as possible components of G. We also examine the situation where an almost simple group G with socle PSL(3,q) acs linetransitively on a linear space. This fits into the wider program of examining those almost simple groups which can act linetransitively on linear spaces, a program motivated by the result in [CP01]. We are able to give strong information about the linetransitive actions of G.
