Right nuclear decomposition of generalized André systems (Q2769466)

Let \(F\) be a finite generalized André system of order \(q\). In the associated translation plane, the right nucleus of \(F\) corresponds to the group \({\mathcal H}_{(0)}\) of all homologies whose axis is the horizontal coordinate axis and whose center is the infinite point of the vertical coordinate axis. Hence there exists a divisor \(d\) of \(q-1\), called the index of \({\mathcal H}_{(0)}\), such that the order of this right nucleus equals \((q-1)/d\). NEWLINENEWLINENEWLINEThe author determines necessary and sufficient numerical conditions for \(q\) and \(d\) such that a generalized André system of order \(q\) and index \(d\) for the homology group \({\mathcal H}_{(0)}\) exists. This yields a vast generalization of known results on the existence of Dickson pairs, which correspond to the case \(d =1\). NEWLINENEWLINENEWLINEThe case \(d=2\) has been investigated by \textit{Y. Hiramine} and \textit{N. L. Johnson} [Geom. Dedicata 41, 175-190 (1992; Zbl 0753.51002)]. However, the results of the present papers show that these authors overlooked one case.