Half nearfield planes (Q1333602)

This paper is a continuation of specific results. [See the author, J. Geom. 42, No. 1/2, 109-125 (1991; Zbl 0754.51006); \textit{Y. Hiramine} and the author, Geom. Dedicata 41, No. 2, 175-190 (1992; Zbl 0753.51002) and ibid. 43, No. 1, 17-33 (1992; Zbl 0764.51009); the author, Simon Stevin 65, No. 3/4, 199-215 (1991; Zbl 0760.51009) inter alia.] The author calls a translation plane \(\pi\) of order \(p^ r\) a half nearfield plane if this plane admits an affine homology group of order \((p^ r - 1)/2\). Not all half nearfield planes are nearfield planes. The author investigates translation planes \(\pi\) of order \(q^ 2\) with spreads in \(\text{PG} (3,q)\) that admit an affine homology group of order \((q^ 2 - 1)/2\). A main result is the following: In every such plane \(\pi\) one of the following possibilities occurs: (i) \(\pi\) is Desarguesian, (ii) \(\pi\) is regular nearfield, (iii) \(\pi\) is the proper half nearfield plane of dimension 2, (iv) the order is \(23^ 2\) and the plane is the irregular nearfield plane, (v) the order is \(7^ 2\) and the plane is the irregular nearfield plane, or (vi) the order is \(7^ 2\) and the plane is the exceptional Lüneburg plane admitting \(\text{SL} (2,3)\). With that is given a classification. The investigation of the paper gives a lot of interesting details, in particular for the cases \(q =23\), 47, or 7.