Abstract
In this article, the question is considered whether there exist finite translation planes with arbitrarily small kernels admitting nonsolvable collineation groups. For any integerN, it is shown that there exist translation planes of dimension >N and orderq 3 admittingGL(2,q) as a collineation group.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
C. Bartolone and T. G. Ostrom. Translation planes of orderq 3 which admitSL(2, q).J. Alg. 99 (1986), 50–57.
P. Dembowski. Finite Geometries. Springer-Verlag, Berlin, New York, Heidelberg, 1968.
J. Fink and M. J. Kallaher. Simple groups acting on translation planes.J. Geom. 29 (1987), 126–139.
D. A. Foulser. Subplanes of partial spreads in translation planes.Bull. London Math. Soc. 4 (1972), 32–38.
D. A. Foulser. Derived translation planes admitting affine elations.Math. Zeit. 131 (1973), 183–188.
D. R. Hughes and F. C. Piper. Projective Planes. Springer-Verlag, Berlin, New York, Heidelberg, 1973.
V. Jha and N. L. Johnson. A characterization of some spreads of orderq 3 that admitGL(2,q) as a collineation group.Hokkaido Math. J. 18, no. 1 (1989), 137–147.
V. Jha and N. L. Johnson. An analog of the Albert-Knuth theorem on the orders of finite semifields, and a complete solution to Cofman's subplane problem.Alg., Groups, Geom. 6 (1989), 1–35.
N. L. Johnson. A note on translation planes of large dimension.J. Geom. 33 (1988), 66–72.
N. L. Johnson and T. G. Ostrom. Inherited groups and kernels of derived translation planes.Euro. J. Comb. 11 (1990), 145–149.
W. M. Kantor, Expanded, sliced and spread spreads.Finite Geometries. Ed. N. L. Johnson, et al., Lecture Notes in Pure and Applied Math.82 (1983), Marcel Dekker, 251–261.
W. M. Kantor. Translation planes of orderq 6 admittingSL(2,q 2).J. Comb. Theory, Ser. A32 no. 2 (1982), p. 299- 302.
H. Lüneburg. Translation planes. Springer-Verlag. Berlin, New York, Heidelberg, 1980.
T. G. Ostrom. Elementary Abelian 2-groups in finite translation planes.Arch. d. Math. 36 (1981), 21–22.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Jha, V., Johnson, N.L. Translation planes of large dimension admitting nonsolvable groups. J Geom 45, 87–104 (1992). https://doi.org/10.1007/BF01225768
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01225768