Planes of Order 10 Do Not Have a Collineation of Order 5

The existence of a projective plane of order 10 remains in doubt. If one does exist it may have only the identity collineation. D. R. Hughes [I, 2] showed that for a plane of order n where y1 = 2 (mod 4) and IZ > 2 the collineation group is of odd order. He also showed that for a plane of order IO the only primes dividing the order of the collineation group could be 3, 5, or 11, that for order 3 there would be 3 or 9 fixed points (also lines) and for 5 exactly one fixed point and one fixed line. Whitesides [9] has eliminated the possibility of a collineation of order Il. She has also [lo] eliminated orders 9,25, and 15, so that the only remaining possible orders are 1, 3, or 5. The main result of this paper is to eliminate the order 5.