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

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(qi−1).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.