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The combinatorial structure of the Hughes plane

Published online by Cambridge University Press:  24 October 2008

T. G. Room
T. G. Room
Affiliation:
University of Sydney, New South Wales, 2006, Australia
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Abstract

The main purpose of this paper is to describe the construction of an incidence table for the Hughes plane of order q2. To do this it is necessary first to construct a table of the same pattern for the Galois plane, and this requires the expression in terms of explicit matrices of the Singer cyclic group of the plane as the composition of cyclic groups of orders q2 + q + 1 and q2 − q + 1. The two tables constructed have some combinatorial properties of possible interest.

In the Galois plane of order q2 there are two types of polarities, one with q2 + 1 singular points on a conic, and the other with q3 + 1 singular points on the Hermitian analogue of a conic. Correspondingly in the Hughes plane there are polarities with ½(q3 + q + 2) and with ½(q3 + 2q2 − q + 2) singular points.

The paper concludes with the complete incidence table for the Hughes plane of order 25, together with the data necessary for its construction.

The Hughes plane and some of its properties are described in Hughes(1), Zappa(2), Rosati(3) and Ostrom(4). The property which enables the Hughes plane to be most easily constructed from the corresponding Galois plane, and which will be used in this paper, is to be found in Room (5).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 68 , Issue 2 , September 1970 , pp. 291 - 301
Copyright
Copyright © Cambridge Philosophical Society 1970

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References

REFERENCES

(1)Hughes, D. R.A class of non-Desarguesian planes. Can. J. Math. 9 (1957), 378–388.CrossRefGoogle Scholar
(2)Zappa, G.Sui gruppi di collineazioni dei piani di Hughes. Boll. Un. Mat. Ital. 12 (1957), 507–516.Google Scholar
(3)Rosati, L. A.I gruppi di collineazioni dei piani di Hughes. Boll. Un. Mat. Ital. 13 (1958), 505–513.Google Scholar
(4)Ostrom, T. G.Semi-translation planes. Trans. Amer. Math. Soc. 111 (1964), 1–18.CrossRefGoogle Scholar
(5)Room, T. G.Veblen–Wedderbum hybrid planes. Proc. Roy. Soc. Ser. A309 (1969), 157–170.Google Scholar