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Involutory collineations of finite planes. (English) Zbl 0599.51010

Satz A: Die Ordnung n einer endlichen projektiven Ebene \({\mathcal P}\) sei von der Form \(n=m^ 2\) mit \(m\equiv 2\) oder 3 (mod 4). Dann enthält jede Kollineationsgruppe der Ordnung 4 von \({\mathcal P}\) eine involutorische Perspektivität. Dies impliziert den folgenden Satz B von J. Fink und M. Kallaher, Simple groups acting on translation planes (to appear): Wenn eine Translationsebene von ungerader Ordnung n eine nichtabelsche einfache Kollineationsgruppe G gestattet, dann ist \(n=m^ 2\) mit \(m\equiv 1 (mod 4)\), und alle Involutionen von G sind Baer- Involutionen. Die Klassifikation der endlichen einfachen Gruppen, die beim Beweis von Satz B benutzt wird, liefert weitere Informationen über G.
Reviewer: Th.Grundhöfer

MSC:

51E15 Finite affine and projective planes (geometric aspects)
51A10 Homomorphism, automorphism and dualities in linear incidence geometry
51A40 Translation planes and spreads in linear incidence geometry
20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures
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References:

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