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Indicator Sets in an Affine Space of any Dimension

  • F. A. Sherk (a1)

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It is well known that a translation plane can be represented in a vector space over a field F where F is a subfield of the kernel of a quasifield which coordinatizes the plane [1; 2; 4, p.220; 10]. If II is a finite translation plane of order qr (q = pn, p any prime), then II may be represented in V2r(q), the vector space of dimension 2r over GF(q), as follows:

(i) The points of II are the vectors in V = V2r(q)

(ii) The lines of II are

(a) A set of qr + 1 mutually disjoint r-dimensional subspaces of V.

(b) All translates of in V.

(iii) Incidence is inclusion.

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References

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1. André, J., Uber nicht-Desarquessche Ebenen mit transitiver. Translationsgruppe, Math. Zei. 60 (1954), 156186.
2. Bruck, R. H. and Bose, R. C., The construction of translation planes from projective spaces, J. Algebr. 1 (1964), 85102.
3. Bruen, A., Spreads and a conjecture of Bruck and Bose, J. Algebr. 23 (1972), 519537.
4. Dembowski, P., Finite geometries (Springer-Verlag, Berlin, 1968).
5. Foulser, D. A., Collineation groups of generalized André planes, Can. J. Math. 21 (1969), 358369.
6. Hering, C., Eine nicht-desarquessche zweifach transitive affine Ebene der Ordnung 27, Abh. Math. Sem. Hamb. 34 (1969), 203208.
7. Lunelburg, H., Die Suzukigruppen und ihre Geometrien (Springer-Verlag, Berlin, 1965).
8. Maduram, D. M., Matrix representation of translation planes, Geometriae Dedicat. 4 (1975), 485492.
9. Ostrom, T. G., A characterization of generalized André planes, Math. Zeit. 110 (1969), 19.
10. Ostrom, T. G., Finite translation planes (Springer-Verlag, Berlin, 1970).
11. Ostrom, T. G., Classification of finite translation planes, Proceedings of the international conference on projective planes held at Washington State University, April 25-28, 1973 (Washington State University Press, 1973).
12. Rao, M. L. N. and Rao, K. K., A new flag transitive affine plane of order 27, Proc. of the Amer. Math. Soc. 59 (1976), 337345.
13. Sherk, F. A. and Pabst, G., Indicator sets, reguli, and new a class of spreads, Can. J. Math. 29 (1977), 132154.
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Indicator Sets in an Affine Space of any Dimension

  • F. A. Sherk (a1)

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