Commutativity in Finite Planes of Type I - 4

This generalizes a result of Hughes [4, 5], who proved Theorem 1 when the plane has order ~ 250. Our proof essentially amounts to formalizing Hughes' approach, and then applying results on the structure of a finite ~ o u p having a nontrivial normal parti t ion [1, 2, 7]. Coordinatizing a plane of type I-4 in the usual manner with U = A, V ---B, and 0 = C (see [3, w 3.1]), we obtain a linear planar ternary ring which has associative multiplication and satisfies both distributive laws. Such a ternary ring is called a planar division neo-ring (PDNR). The multiplicative group of a P D N R is isomorphic to the group of all (A, B C)-homologies. Thus, Theorem 1 will follow from the following generalization of Wedderburn's Theorem.