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On a C–plane of order 25

Published online by Cambridge University Press:  17 April 2009

M.L. Narayana Rao  and
K. Satyanarayana
M.L. Narayana Rao
Affiliation:
Department of Mathematics, Osmania University, Hyderabad–500 007, India
K. Satyanarayana
Affiliation:
Department of Mathematics, Osmania University, Hyderabad–500 007, India
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Abstract

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Rao, Rodabaugh, Wilke and Zemmer [J. Combin. Theory Ser. A. 11 (1971), 72–92] constructed a number of new VW systems called C-systems from the exceptional near–fields and established that they coordinatize translation planes not isomorphic to generalized André planes. In this paper the translation complement of the plane coordinatized by the C-system I–1 has been found. This plane has the interesting property that its translation complement divides the ideal points into two orbits of lengths 10 and 16. Further, the translation complement contains a subgroup isomorphic to SL(2,5) and therefore one of the exceptional Walker's planes of order 25 [H. Luneberg, Translation Planes, Springer-Verlag (1980), pp.235–244] is indeed the C–plane corresponding to the C–system I–1, which was discovered in 1969.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 30 , Issue 1 , August 1984 , pp. 27 - 36
Copyright
Copyright © Australian Mathematical Society 1984

References

[1]Dembowski, P., Finite Geometries, (Springer-Verlag, Berlin, 1968).CrossRefGoogle Scholar
[2]Foulser, D.A., “Solvable flag transitive affine planes”. Math. Z. 86 (1964), 191–204.CrossRefGoogle Scholar
[3]Lueder, Kenneth, “Derivability in the irregular nearfield planes”. Proceedings of the International Conference on projective planes, Washington State University Press, 1973, 181–189.Google Scholar
[4]Luneberg, H., Translation planes, (Springer-Verlag, 1980).CrossRefGoogle Scholar
[5]Hall, Marshall, The Theory of Groups, (Macmillan, New York, 1959).Google Scholar
[6]Rao, M.L. Narayana and Rao, K. Kuppu Swamy, “Full collineation group of Foulser's flag transitive plane of order 25”, Houston J. Math. 7 (1981), 115–120.Google Scholar
[7]Rao, M.L. Narayana, Roadabaugh, D.J., Wilke, F.W. and Zemmer, J.L., “A new class of finite translation planes obtained from the exceptional near–fields”, J. Combin. Theory Ser. A. 11 (1971), 72–92.CrossRefGoogle Scholar
[8]Rao, M.L. Narayana and Satyanarayana, K., “Collineations in the translation complement fixing an ideal point of a translation plane”, to appear.Google Scholar
[9]Ostrom, T., “Classification of finite translation planes”, Proceedings of the International Conference on projective planes, Washington State University Press, 1973, 195–213.Google Scholar
[10]Ostrom, T.G., “Elementary abelian 2-groups on the line at infinity of translation planes”, J. Geom. 17 (1981), 128–139.CrossRefGoogle Scholar
[11]Satyanarayana, K. and Rao, K. Kuppu Swamy, “Full collineation group of the second of Foulser's flag transitive planes of order 25”, Houston J. Math. 7 (1981), 537–543.Google Scholar
[12]Walker, M., “A class of translation planes”, Geom Dedicata 5 (1976) 135–146.CrossRefGoogle Scholar