Beitr\ EMIS ELibM Electronic Journals Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
Vol. 44, No. 1, pp. 25-46 (2003)

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RWPRI and (2T)$_1$ Flag-transitive Linear Spaces

F. Buekenhout, P.-O. Dehaye and D. Leemans

Département de Mathématique, Université Libre de Bruxelles, C.P.216 - Géométrie, Boulevard du Triomphe, B-1050 Bruxelles, Belgium, e-mail:; [address of Dehaye:] Department of Mathematics, Building 380, Stanford University, 94305-2125 Stanford, CA, U.S.A.

Abstract: The classification of finite flag-transitive linear spaces is almost complete. For the thick case, this result was announced by Buekenhout, Delandtsheer, Doyen, Kleidman, Liebeck and Saxl, and in the thin case (where the lines have 2 points), it amounts to the classification of $2$-transitive groups, which is generally considered to follow from the classification of finite simple groups. These two classifications actually leave an open case, which is the so-called $1$-dimensional case. In this paper, we work with two additional assumptions. These two conditions, namely (2T)$_1$ and RWPRI, are taken from another field of study in Incidence Geometry and allow us to obtain a complete classification, which we present at the end of this paper. In particular, for the $1$-dimensional case, we show that the only (2T)$_1$ flag-transitive linear spaces are ${AG}(2,2)$ and ${AG}(2,4)$, with $A\Gamma L (1,4)$ and $A\Gamma L (1,16)$ as respective automorphism groups.

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Electronic version published on: 3 Apr 2003. This page was last modified: 4 May 2006.

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