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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.

MSC:
51A40 Translation planes and spreads in linear incidence geometry
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[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
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