The linearity question for Abelian groups on translation planes (Q1102535)

Let \(\pi\) be a translation plane of order \(p^{2r}=q^ 2\), p a prime, r an integer, with kernel GF(p). Suppose that \(\pi\) admits an Abelian collineation group \({\mathcal G}\) of order \(q^ 2\) in the translation complement of \(\pi\). The authors prove the following: (i) \({\mathcal G}\) is in the linear translation complement. (ii) If q is even then \(\pi\) is a semifield plane or a ``Betten plane''. (iii) If q is odd then \(\pi\) is a semifield plane or a ``desirable plane''. [Cf. the second author and \textit{F. W. Wilke}, Geom. Dedicata 15, 293-312 (1984; Zbl 0532.51005)].