A <i>linear space </i>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 <i>G </i>which is transitive upon the set of lines of the space. Within the set of all linear spaces lies a particularly important subset: the <i>projective planes. </i>Results exist in the literature [Cam04, CP93] classifying the possible minimal normal subgroups of a group <i>G </i>acting line-transitively on a finite projective plane. We rewrite some of these results to deal with <i>components</i> rather than with minimal normal subgroups. We then prove that, if a group <i>G</i> acts on a projective plane which is not Desarguesian, the <i>G</i> 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 <i>G. </i> We also examine the situation where an almost simple group <i>G </i>with socle <i>PSL</i>(3,<i>q</i>) acs line-transitively on a linear space. This fits into the wider program of examining those almost simple groups which can act line-transitively on linear spaces, a program motivated by the result in [CP01]. We are able to give strong information about the line-transitive actions of <i>G</i>.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:599416 |
| Date | January 2005 |
| Creators | Gill, N. P. |
| Publisher | University of Cambridge |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
Page generated in 0.0018 seconds