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International Journal of Mathematics and Mathematical Sciences
Volume 3 (1980), Issue 4, Pages 675-694
http://dx.doi.org/10.1155/S0161171280000488

Translation planes of even order in which the dimension has only one odd factor

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

Received 2 January 1980

Copyright © 1980 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

Let G be an irreducible subgroup of the linear translation complement of a finite translation plane of order qd where q is a power of 2. GF(q) is in the kernel and d=2sr where r is an odd prime. A prime factor of |G| must divide (qd+1)i=1d(qi1).

One possibility (there are no known examples) is that G has a normal subgroup W which is a W-group for some prime W.

The maximal normal subgroup 0(G) satisfies one of the following:

1. Cyclic. 2. Normal cyclic subgroup of index r and the nonfixed-point-free elements in 0(G) have order r. 3. 0(G) contains a group W as above.