On the Lorimer-Rahilly and Johnson-Walker translation planes

Abstract

We investigate finite translation planes of dimension d over the kernel K=GF(q), where q=p k with p a prime, having a collineation group G with either G=PSL (2, w) or G=SL(3, w) 9 where w is a prime power. We derive several restrictions on the planes; for example, if p is odd then 4 divides d. We also give a new characterization of the Lorimer-Rahilly and Johnson-Walker planes of order 16, which is more general than that of Johnson and Ostrom. In addition, we give many examples indicating how good are our results.