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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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##### References:
  A.Caggegi and A.Herzer, The Generalized André systemsA (F,Γ, (g_{$$i$$})$$, (f\_{}\{i\}),\textit{ε)}$$. To appear, Abh. Math. Sem. Hamburg58 (1989).  A.Hasse, Zahlentheorie. Berlin 1969.  H.Lüneburg, Translation Planes. Berlin-Heidelberg-New York 1980. · Zbl 0446.51003  R. Rink, Eine Klasse unendlicher verallgemeinerter André-Ebenen. Geom. Dedicata6, 55-80 (1977). · Zbl 0363.50013
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