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On homologies in finite combinatorial geometries

Published online by Cambridge University Press:  17 April 2009

P.B. Kirkpatrick
P.B. Kirkpatrick
Affiliation:
Department of Pure Mathematics, University of Sydney, Sydney, New South Wales.
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Abstract

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Any subset π* of the set of all planes through a line in a finite projective space PG(m, q) determines a subgeometry G(π*) of the combinatorial geometry associated with PG(m, q). In this paper the geometries G(π*) of rank greater than three in which every line contains at least four points, are characterized in terms of the existence of a certain set of automorphism groups Γ(C, X); where X is a copoint and C a point not in X, and each non-trivial element of Γ(C, X) fixes X and every copoint through C and fixes C and every point in X, but no other point; and where Γ(C, X) acts transitively on the points distinct from C and not in X of some line through C. As a corollary of the main theorem we obtain a characterization of the finite projective spaces PG(m, q) with m ≥ 3 and q ≥ 3.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 13 , Issue 1 , August 1975 , pp. 85 - 99
Copyright
Copyright © Australian Mathematical Society 1975

References

[1]André, Johannes, “Über Perspektivitäten in endlichen projektiven Ebenen”, Arch. Math. 6 (1954), 29–32.Google Scholar
[2]Barlotti, Adriano, “Le possibili configurazioni del sistema delle coppie punto-retta (A, α) per cui un piano grafico risulta (A, α)-transitivo”, Boll. Un. Mat. Ital. (3) 12 (1957), 212–226.Google Scholar
[3]Crapo, Henry H. and Rota, Gian-Carlo, On the foundations of combinatorial theory: combinatorial geometries, Preliminary edition (MIT Press, Cambridge, Massachusetts; London, England; 1970).Google Scholar
[4]Dembowski, P., Finite geometries (Ergebnisse der Mathematik und ihrer Grenzgebiete, 44. Springer-Verlag, Berlin, Heidelberg, New York, 1968).Google Scholar
[5]Gleason, Andrew M., “Finite Fano planes”, Amer. J. Math. 78 (1956), 797–807.CrossRefGoogle Scholar
[6]Lenz, Hanfried, “Kleiner Desarguesscher Satz und Dualität in projektiven Ebenen”, Jber. Deutsch. Math.-Verein 57 (1954), 20–31.Google Scholar
[7]Wielandt, Helmut, Finite permutation groups (Academic Press, New York and London, 1964).Google Scholar