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Finite Projective Geometries

  • Gerald Berman (a1)

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James Singer [12] has shown that there exists a collineation which is transitive on the (t - 1)-spaces, that is, (t - 1)-dimensional linear subspaces, of PG(t, p n). In this paper we shall generalize this result showing that there exist t - r collineations which together are transitive on the s-spaces of PG(t, p n). An explicit construction will be given for such a set of collineations with the aid of primitive elements of Galois fields. This leads to a calculus for the linear subspaces of finite projective geometries.

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References

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1. Bussey, W. H., Tables of Galois fields, Bull. Amer. Math. Soc, vol. 12 (1905-6), 2238.
2. Bussey, W. H. Tables of Galois fields, Bull. Amer. Math. Soc, vol. 16 (1909-10), 188206.
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10. Rao, C. R., Difference sets and combinatorial arrangements derivable from finite geometries, Proc. Nat. Inst. Sci. India (1946), 123135.
11. Robinson, G. de B., The foundations of geometry (Toronto, 1940).
12. Singer, J., A theorem in finite projective geometry and some applications to number theory, Trans. Amer. Math. Soc, vol. 43 (1938), 377385.
13. Snapper, E., Periodic linear transformations of affine and projective geometries, Can. J. Math., vol. 2 (1950), 149151.
14. Sommerville, D. M. Y., An introduction to the geometry of n dimensions (London, 1929).
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16. Veblen, O. and Bussey, W. H., Finite projective geometries, Trans. Amer. Math. Soc, vol. 7 (1906), 244.
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Finite Projective Geometries

  • Gerald Berman (a1)

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