zbMATH — the first resource for mathematics

Zu einem Satz von H. Lüneburg über verallgemeinerte André-Ebenen. (About a theorem of H. Lüneburg on generalized André planes.). (German) Zbl 0633.51003
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.) [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 \(\Lambda\) [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.

51A40 Translation planes and spreads in linear incidence geometry
Full Text: DOI
[1] A.Caggegi and A.Herzer, The Generalized André systemsA (F,Γ, (g_{\(i\)})\(, (f\_{}\{i\}),\textit{ε)}\). To appear, Abh. Math. Sem. Hamburg58 (1989).
[2] A.Hasse, Zahlentheorie. Berlin 1969.
[3] H.Lüneburg, Translation Planes. Berlin-Heidelberg-New York 1980. · Zbl 0446.51003
[4] R. Rink, Eine Klasse unendlicher verallgemeinerter André-Ebenen. Geom. Dedicata6, 55-80 (1977). · Zbl 0363.50013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.