International Journal of Mathematics and Mathematical Sciences
Volume 7 (1984), Issue 2, Pages 327-338

On rank 5 projective planes

Otto Bachmann

Département de mathématiques, Ecole polytechnique fédérale, Lausanne CH-1015, Swaziland

Received 29 December 1983; Revised 16 April 1984

Copyright © 1984 Otto Bachmann. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


In this paper we continue the study of projective planes which admit collineation groups of low rank (Kallaher [1] and Bachmann [2,3]). A rank 5 collineation group of a projective plane of order n3 is proved to be flag-transitive. As in the rank 3 and rank 4 case this implies that is not desarguesian and that n is (a prime power) of the form m4 if m is odd and n=m2 with m0mod4 if n is even. Our proof relies on the classification of all doubly transitive groups of finite degree (which follows from the classification of all finite simple groups).