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A generalisation of von Staudt’s theorem on cross-ratios

Part of: Projective and enumerative geometry Permutation groups Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  22 February 2019

YATIR HALEVI  and
ITAY KAPLAN
YATIR HALEVI*
Affiliation:
Einstein Institute of Mathematics, Edmond J. Safra Campus, The Hebrew University of Jerusalem, Givat Ram 91904, Jerusalem, Israel
ITAY KAPLAN
Affiliation:
Einstein Institute of Mathematics, Edmond J. Safra Campus, The Hebrew University of Jerusalem, Givat Ram 91904, Jerusalem, Israel
*
e-mails: yatir.halevi@mail.huji.ac.il, kaplan@math.huji.ac.il
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Abstract

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A generalisation of von Staudt’s theorem that every permutation of the projective line that preserves harmonic quadruples is a projective semilinear map is given. It is then concluded that any proper supergroup of permutations of the projective semilinear group over an algebraically closed field of transcendence degree at least 1 is 4-transitive.

MSC classification

Primary: 14N99: None of the above, but in this section
Secondary: 20B27: Infinite automorphism groups 20E28: Maximal subgroups
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 168 , Issue 3 , May 2020 , pp. 601 - 612
Copyright
Copyright © Cambridge Philosophical Society 2019

Footnotes

†

Supported by the European Research Council grant 338821, by ISF grant No. 181/16 and ISF grant No. 1382/15.

‡

Supported by the Israel Science Foundation grants no. 1533/14 and 1254/18.

References

REFERENCES

Beutelspacher, A. and Rosenbaum, U.. Projective geometry: from foundations to applications (Cambridge University Press, Cambridge, 1998).Google Scholar
Bogomolov, F. and Rovinsky, M.. Collineation group as a subgroup of the symmetric group. Cent. Eur. J. Math., 11(1) (2013), 17–26.Google Scholar
Coolidge, J. L.. The rise and fall of projective geometry. Amer. Math. Monthly 41(4):1934, 217–228.CrossRefGoogle Scholar
Havlicek, H.. Von Staudt’s theorem revisited. Aequationes Math. 89(3) (2015), 459–472.CrossRefGoogle Scholar
Herzer, A.. Chain geometries. In Handbook of Incidence Geometry, pages 781–842 (North-Holland, Amsterdam, 1995).CrossRefGoogle Scholar
Hoffman, A. J.. A note on cross ratio. Amer. Math. Monthly 58 (1951), 613–614.Google Scholar
Hua, L.-K.. Some properties of a sfield. Proc. Natl. Acad. Sci., 35(9) (1949), 533–537.CrossRefGoogle ScholarPubMed
Jacobson, N.. Structure and representations of Jordan algebras. Amer. Math. Soc. Colloq. Publ., vol. XXXIX (American Mathematical Society, Providence, R.I., 1968).Google Scholar
Kaplan, I. and Simon, P.. The affine and projective groups are maximal. Trans. Amer. Math. Soc, 368(7) (2016), 5229–5245.CrossRefGoogle Scholar
Schreier, O. and Sperner, E.. Einführung in die Analytische Geometrie und Algebra, volume 2 (BG Teubner, 1935).Google Scholar
Voelke, J.-D.. Le théorème fondamental de la géométrie projective: évolution de sa preuve entre 1847 et 1900. Arch. Hist. Exact Sci. 62(3) (2008), 243–296.CrossRefGoogle Scholar
von Staudt, K. G. C.. Geometrie der Lage (Verlag von Bauer und Raspe (Julius Merz), Nürnberg, 1847).Google Scholar