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Flag-transitive collineation groups of finite projective spaces

1962; Duke University Press; Volume: 6; Issue: 3 Linguagem: Inglês

10.1215/ijm/1255632503

1945-6581

D. G. Higman,

Rings, Modules, and Algebras

A flag in a projective space (P of dimension d _>-2 is a sequence So c S c c S_ of linear subvarieties of ( such that Si has dimension i (i 0, 1, d 1)-Thus, for example, a flag in a projective plane is an incident point-line pair.A collineation group G of ( will be called flag-transitive if any one flag can be carried onto any other by some collineation in G.The little projective group of a Desarguesian (i.e., the group generated by all elations of , isomorphic with PSL+(F), where F is the coordinatizing field) is flag-transitive.Thus the following theorem can be considered as giving a geometric characteriza- tion of PSL+(F) for finite F. If the number of points on each line of is n + 1 we will refer to n as the order of .(For d > 2 this differs from the order of the symmetric design formed by the points an hyperplanes of .)THEOREM.A flag-transitive collineation group G of a Desarguesian projective spe of dimension d 2 and finite order n must contain the little projective group of unless () d 2, n 2, and ]GJ 3.7, or (b) d 2, n 8, and G 9.73, or (c) d 3, n 2, and G is isomorphic with the alternating group A of degree 7.For d 2 this theorem coincides with Theorem 1 of [6].The extension to dimensions 3 (where, of course, the Desarguesian property necessarily holds) is obtained in this paper as an application of extensions of results of Andr [1], Gleason [5], and Wagner [9] concerning perspectivities, together with a special result about embeddings of PSL(F) in PGL+z(F).The exceptions stated in the theorem are real.In case (c), G A is doubly transitive on the points of .Concerning the question whether the Desarguesian condition can be moved from the hypotheses to the conclusion of the theorem, i.e., whether the existence of a flag-transitive collineation group on a finite projective plane implies Desargues' Theorem, see [6].