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A note on planes of characteristic three

Published online by Cambridge University Press:  17 April 2009

Michael J. Kallaher
Affiliation:
Department of Pure and Applied Mathematics, Washington State University, Pullman, Washington, USA.
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Abstract

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A projective plane has characteristic three if in every ternary ring coordinatizing it all elements ≠ 0 of the additive loop have order 3. We show that if a finite plane of characteristic three is coordinatized by a near-field, then the plane is desarguesian.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1972

References

Rererences

[1]Dembowski, P., Finite geometries (Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 44. Springer-Verlag, Berlin, Heidelberg, Hew York, 1968).CrossRefGoogle Scholar
[2]Foulser, David A., “A generalization of André's systems”, Math. Z. 100 (1967), 380395.CrossRefGoogle Scholar
[3]Gleason, Andrew M., “Finite Fano planes”, Amer. J. Math. 78 (1956), 797807.CrossRefGoogle Scholar
[4]Hall, Marshall Jr, Combinatorial theory (Blaisdell Publishing Co. [Ginn and Co.], Waltham, Massachusetts; Toronto, Ontario; London; 1967).Google Scholar
[5]Keedwell, A.D., “On the order of projective planes with characteristic”, Rend. Mat. e Appl. (5) 22 (1963), 498530.Google Scholar
[6]Keedwell, A.D., “A class of configurations associated with projective planes with characteristic”, Arch. Math. 15 (1964), 470480.CrossRefGoogle Scholar
[7]Lombardo-Radice, Lucio, “Sul rango dei piani grafici finite a caratteristica 3”, Boll. Un. Mat. Ital. (3) 10 (1955), 172177.Google Scholar
[8]Lombardo-Radice, Lucio, “Su alcuni caratteri dei piani grafici”, Rend. Sem. Mat. Univ. Padova 24 (1955), 312345.Google Scholar
[9]Zappa, Guido, “Piani grafici a caratteristica 3”, Ann. Mat. Pura Appl. (4) 49 (1960), 157166.CrossRefGoogle Scholar