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International Journal of Mathematics and Mathematical Sciences
Volume 2 (1979), Issue 2, Pages 187-208
http://dx.doi.org/10.1155/S0161171279000181

Translation planes of odd order and odd dimension

Department of Pure and Applied Mathematics, Washington State University, Pullman 99164, Washington, USA

Received 8 December 1978

Copyright © 1979 Hindawi Publishing Corporation. 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.

Abstract

The author considers one of the main problems in finite translation planes to be the identification of the abstract groups which can act as collineation groups and how those groups can act.

The paper is concerned with the case where the plane is defined on a vector space of dimension 2d over GF(q), where q and d are odd. If the stabilizer of the zero vector is non-solvable, let G0 be a minimal normal non-solvable subgroup. We suspect that G0 must be isomorphic to some SL(2,u) or homomorphic to A6 or A7. Our main result is that this is the case when d is the product of distinct primes.

The results depend heavily on the Gorenstein-Walter determination of finite groups having dihedral Sylow 2-groups when d and q are both odd. The methods and results overlap those in a joint paper by Kallaher and the author which is to appear in Geometriae Dedicata. The only known example (besides Desarguesian planes) is Hering's plane of order 27 (i.e., d and q are both equal to 3) which admits SL(2,13).