OJDM  Vol.2 No.1 , January 2012
On Transposed Translation Planes
This paper is devoted to the study of a translation plane π(C) associated with a t-spread set C and its transposed t-spread set C t. In this paper, an explicit matrix form of the inverse of an isomorphism from a translation plane into another translation plane associated with t-spread sets is derived and proved that two translation planes associated with t-spread sets are isomorphic if and only if their corresponding transposed translation planes are isomorphic. Further, it is shown that the transpose of a flag-transitive plane is flag-transitive and derived a necessary and sufficient condition for a translation plane π(C) to be isomorphic to its transposed translation plane.

Cite this paper
K. Satyanarayana and K. Kameswari, "On Transposed Translation Planes," Open Journal of Discrete Mathematics, Vol. 2 No. 1, 2012, pp. 35-43. doi: 10.4236/ojdm.2012.21007.
[1]   P. Dembowski, “Finite Geometries,” Springer-Verlag, New York, 1997.

[2]   R. H. Bruck and R. C. Bose, “The Construction of Translation Planes from Projective Spaces,” Journal of Algebra, Vol. 1, No. 1, 1964, pp. 85-102. doi.org/10.1016/0021-8693(64)90010-9

[3]   R. H. Bruck and R. C. Bose, “Linear Representations of Projective Planes from Projective Spaces,” Journal of Algebra, Vol. 4, No. 1, 1966, pp. 117-172. doi.org/10.1016/0021-8693(66)90054-8

[4]   F. A. Sherk, “Indicator Sets in Affine Planes of Any Dimension,” Canadian Journal of Mathematics, Vol. 31, 1979, pp. 211-224. doi.org/10.4153/CJM-1979-022-6

[5]   D. M. Maduram, “Matrix Representation of Translation Planes,” Geometriae Dedicata, Vol. 4, No. 2-4, 1975, pp. 485-492. doi.org/10.1007/BF00148776

[6]   M. L. Narayana Rao and E. H. Davis, “Construction of Translation Planes from T-Spread Sets,” Journal of Combinatorial Theory, Series A, Vol. 14, No. 2, 1973, pp. 201-208. doi.org/10.1016/0097-3165(73)90021-6

[7]   D. M. Maduram, “Transposed Translation Planes,” Proceedings of American Mathematical Society, Vol. 53, 1975, pp. 265-270. doi.org/10.1090/S0002-9939-1975-0383237-5

[8]   J. Andre, “Uber Nicht-Desarguessche Ebenen Mit Trassittver Translation Gruppe,” Mathematische Zeitschrift, Vol. 60, No. 1, 1954, pp. 156-186. doi.org/10.1007/BF01187370

[9]   D. A. Foulser, “Solvable Flag-Transitive Affine-Groups,” Mathematische Zeitschrift, Vol. 86, No. 3, 1964, pp. 191- 204. doi.org/10.1007/BF01110390

[10]   A. Wagner, “On Finite Affine Line Transitive Planes,” Mathematische Zeitschrift, Vol. 87, No. 1, 1965, pp. 1-11. doi.org/10.1007/BF01109922