Zu einem Satz von H. Lüneburg über verallgemeinerte André-Ebenen. (About a theorem of H. Lüneburg on generalized André planes.) (Q1096170)

Lüneburg's theorem mentioned here states a property (L) of the collineation group of a translation plane \(\pi\) which forces \(\pi\) to be coordinatized by a generalized André system (g.A.s.) [\textit{H. Lüneburg}, Translation planes (1980; Zbl 0446.51003), see Theorem 12.1]. To go into the converse direction, here g.A.s.'s A(F,\(\Gamma\),f) are considered, where \(\Gamma\) is a finite Galois group for a field extension F/K and \(f: F^ *\to \Gamma\) defines the multiplication \(\circ\) by \(a\circ b=ab^{f(a)}\). A condition (B) is stated for a Galois extension F/K with finite Galois group \(\Gamma\) that for any g.A.s. \(A=A(F,\Gamma,f)\) the translation plane coordinatized by A has property (L). Necessarily the Sylow subgroups of \(\Gamma\) then are cyclic or generalized quaternion. Three examples are given where (B) is fulfilled: (i) \(\Gamma\) a cyclic p-group (or for \(p=2\) generalized quaternion group), (ii) certain field extensions by roots of unity including the finite g.A.S.'s and the translation planes of type \(\Lambda\) [\textit{R. Rink}, Geom. Dedicata 6, 55-79 (1977; Zbl 0363.50013)], (iii) the Kummer extensions, i.e \(\Gamma\) cyclic of order d, \(char K \nmid d,\) and K contains the d-th roots of unity.