A characterization of the Hall planes by planar and nonplanar involutions (Q1263812)

The author considers translation planes of order \(q^ 2\) which admit a Baer group of order q and \(\geq 1+q\) involutory control collineations. For q odd, other, non Hall translation planes are known, not possessing this configuration of groups. However, for even q these Hall planes can be characterized as follows: Let \(\pi\) be a translation plane of even order \(q^ 2\) which admits a Baer collineation group of order q and \(\geq 1+q\) nontrivial elations (all groups are assumed to be in the translation complement). Then \(\pi\) is the Hall plane of order \(q^ 2\), and conversely, the Hall plane admits such groups.