Abstract
The first examples of groups of projectivities in non-des- arguesian planes were computed by A. Barlotti [1959], He showed that the group of projectivities of the three known non-desar- guesian planes of order 9 is in fact the symmetric group of degree 10 and that the group of projectivities of the Hall plane of order 16 contains the alternating group of degree 17. Later on he showed that this group is actually A17. In the sequel A. Herzer, J. Joussen, and A. Longwitz determined the group of projectivities for several infinite classes of projective planes (see the bibliography for bibliographical details) showing that it always contains the alternating group of degree q + 1, where q is the order of the plane under consideration. Almost all their results are included in the results by Th. Grundhöfer which will be presented here for the first time. I would like to thank Th. Grundhöfer very much indeed for allowing me to incorporate his material into this note.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Bibliography
A. Barlotti, La determinazione del gruppo delle proiettività di una retta in sè in alcuni particolari piani grafici finiti non-desarguesiani. Boll. Un. Mat. III. Ser. 14, (1959), 543–547.
H. Davenport & H. Hasse, Die Nullstellen der Kongruenzzetafunktion in gewissen zyklischen Fällen. J. r. a. Math. 172, (1935), 151–182.
Ch. Hering, Transitive Linear Groups and Linear Groups which Contain Irreducible Subgroups of Prime Order. Geom. Ded. 2, (1974), 425–460.
A. Herzer, Über die Gruppe der Projektivitäten einer Geraden auf sich in einigen endlichen Fastkörperebenen. Geom. Ded. 1, (1972), 47–64.
A. Herzer, Dualitäten mit zwei Geraden aus absoluten Punkten in projektiven Ebenen. Math. Z. 129, (1972a), 235–257.
A. Herzer, Über die Gruppe der Projektivitäten einer Geraden auf sich in einigen endlichen Translationsebenen. Geom. Ded. 2, (1973), 363–385.
A. Herzer, Die Gruppe Л (g) in den endlichen Hall-Ebenen. Geom. Ded. 2, (1973), 1–12.
A. Herzer, Die Gruppe Л (g) in den endlichen Andrè-Ebenen gerader Ordnung. Geom. Ded. 3, (1974), 241–249.
D. Holt, Triply Transitive Permutation Groups in which an Involution Central in a Sylow 2-Subgroup Fixes a Unique Point. J. Lond. Math. Soc. (2) 15, (1977), 55–65.
J. Joussen, Zum Transitivitätsverhalten der Projektivitätengruppe einer endlichen Fastkörperebene. Abh. Math. Sem. Hamb. 35, (1971), 230–241.
A. Longwitz, Die Gruppe der Projektivitäten einer Geraden auf sich in Andrè-Ebenen vom Grad 2. Geom. Ded. 4, (1975), 263–270.
A. Longwitz, Die Gruppe der Projektivitäten einer Geraden auf sich in endlichen André-Ebenen. Geom. Ded. 8, (1979), 501–511.
H. Lüneburg, An Axiomatic Treatment of Ratios in an Affine Plane. Arch. Math. 18, (1967), 444–448.
H. Lüneburg, Lectures on Projective Planes. Chicago 1969.
H. Lüneburg, Transitive Erweiterungen endlicher Permutationsgruppen. Berlin-Heidelberg-New York 1969a.
H. Lüneburg, Translation Planes. Berlin-Heidelberg-New York 1980.
H. Lüneburg, Grundlagen der ebenen Geometrie. 7 Studienbriefe. Hagen 1980a.
B. Mortimer, Permutation Groups Containing Affine Groups of the Same Degree. J. Lond. Math. Soc. (2) 15, (1977), 445–455.
A. Schleiermacher, Über projektive Ebenen, in denen jede Projektivität mit sechs Fixpunkten die Identität ist. Math. Z. 123, (1971), 325–339.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1981 D. Reidel Publishing Company
About this paper
Cite this paper
Lüneburg, H. (1981). Some New Results on Groups of Projectivities. In: Plaumann, P., Strambach, K. (eds) Geometry — von Staudt’s Point of View. NATO Advanced Study Institutes Series, vol 70. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-8489-9_9
Download citation
DOI: https://doi.org/10.1007/978-94-009-8489-9_9
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-009-8491-2
Online ISBN: 978-94-009-8489-9
eBook Packages: Springer Book Archive