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International Journal of Mathematics and Mathematical Sciences
Volume 4 (1981), Issue 4, Pages 711-724
http://dx.doi.org/10.1155/S0161171281000549

Collineation groups of translation planes of small dimension

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

Received 3 September 1980; Revised 30 March 1981

Copyright © 1981 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

A subgroup of the linear translation complement of a translation plane is geometrically irreducible if it has no invariant lines or subplanes. A similar definition can be given for “geometrically primitive”. If a group is geometrically primitive and solvable then it is fixed point free or metacyclic or has a normal subgroup of order w2a+b where wa divides the dimension of the vector space. Similar conditions hold for solvable normal subgroups of geometrically primitive nonsolvable groups. When the dimension of the vector space is small there are restrictions on the group which might possibly be in the translation complement. We look at the situation for certain orders of the plane.