Maximal Baer groups in translation planes and compatibility with homology groups (Q1906235)

The authors determine all translation planes of order \(q^2\) and kernel containing \(GF(q)\) which admit at least two Baer groups of order \(q - 1\) in the translation complement: Such a plane \(\pi\) is in general a Hall plane. There are, however, some exceptional cases for \(q = 3,4,5\): the Desarguesian plane of order 9, the semifield plane of order 16 with kernel \(GF(4)\), the irregular nearfield plane of order 25, the exceptional Walker plane of order 25 with orbit structure 10, 16 on the line at infinity. Moreover, due to a result by one of the authors [\textit{N. L. Johnson}, Simon Stevin 63, No. 2, 167-188 (1989; Zbl 0702.51006)] the results are transferred to partial hyperbolic flocks: thus some new examples of maximal partial hyperbolic flocks in \(PG(3,5)\) are constructed.