Two-dimensional flag-transitive planes revisited
A longstanding problem in the field of finite affine planes is the determination of all finite flag-transitive affine planes. By the famous theorem of \textit{A. Wagner} [Math. Z. 87, 1-11 (1965; Zbl 0131.36804)] all such planes are translation planes. Several examples of nondesarguesian flag-transitive affine planes have been discovered over the years including a class by the authors [Geom. Dedicata 27, No. 1, 9-14 (1988; Zbl 0646.51005)] and two classes by \textit{W. Kantor} [Geom. Dedicata 41, No. 2, 191-200 (1992; Zbl 0748.51004)] and \textit{C. Suetake} [Graphs Comb. 7, No. 2, 183-193 (1991; Zbl 0735.51005)]. The authors show that the planes in one class of Kantor-Suetake having dimension 2 over their kernel are just the planes discovered by them. Furthermore, the authors extend a result of Kantor's on the number of isomorphism classes of flag-transitive affine planes. More specifically, if \(q=p^e\) with \(p\) an odd prime, then the number of flag-transitive affine planes having dimension 2 over its kernel \(GF(q)\) is at least \((q-1)/2e\). Combined with an earlier result of the authors this says the number of isomorphism classes when \(q=p\) is exactly \((q-1)/2\), and they can be obtained using the authors' construction method. Also, this implies that the smallest possible order for a two-dimensional finite flag-transitive affine plane not yet discovered is \(3^{10}\).
scientific article; zbMATH DE number 942736