On the Lorimer-Rahilly and Johnson-Walker translation planes

Vikram Jha and Michael Joseph Kallaher

Vol. 103 (1982), No. 2, 409–427

DOI: 10.2140/pjm.1982.103.409

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), 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.

Mathematical Subject Classification 2000

Primary: 51E15

Secondary: 51A40

Milestones

Received: 5 August 1980

Revised: 25 February 1981

Published: 1 December 1982