Abstract
Sufficient and necessary conditions have been obtained for the following: (1) the substructure formed by a member of the partition of points and a member of the partition of lines to be a subplane; (2) the centralizer of a multiplier to be a Baer subplane. We establish the cyclicity of a Sylow 3-subgroup of the multiplier group of an abelian Singer group of square planar order. Sufficient conditions for the existence of a Type II divisor of a Singer group are given. For a Singer group of orderpq, p<q, we prove that if the order of the multiplier group is divisible byp, then the plane will admit a cyclic Singer group.
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Baumert, L. D.:Cyclic Difference Sets, Lecture Notes in Math., Springer-Verlag, New York, 1971.
Dembowski, P.:Finite Geometries, Springer, New York, 1968.
Dembowski, P.: Verallgemeinerugen von Transitivitatsklassen endlicher projektiver Ebenen,Math. Z. 69 (1958), 59–89.
Feit, W. and Thompson, J. G.: Solvability of groups of odd order,Pacific J. Math. 13 (1963), 755–1029.
Gorenstein, D.:Finite Groups, Harper and Row, New York, 1968.
Hall, M. Jr.: Cyclic projective planes,Duke Math. J. 14 (1947), 1079–1090.
Ho, C. Y.: On multiplier group of finite cyclic planes,J. Algebra (1989), 250–259.
Ho, C. Y.: Some remarks on order of projective planes, planar difference sets and multipliers,Designs, Codes and Cryptography 1 (1991), 69–75.
Ho, C. Y.: Projective planes with a regular collineation group and a question about powers of a prime,J. Algebra 154 (1993), 141–151.
Ho, C. Y.: On bounds for groups of multipliers of planar difference sets,J. Algebra 148 (1992), 325–336.
Ho, C. Y.: Planar Singer groups with even order multiplier groups,Finite Geometry and Combinatorics, Lecture Notes 191 (1993), pp. 187–198.
Ho, C. Y.: Some basic properties of planar Singer groups, to appear inGeom. Dedicata.
Ho, C. Y. and Pot, A.: On multiplier groups of planar difference sets and a theorem of Kantor,Proc. Amer. Math. Soc. 109 (1990), 803–808.
Hughes D. and Piper, F.:Projective Planes, Springer, New York, 1973.
Jungnickel, D.:Contemporary Design Theory: A Collection of Surveys H. Dinitz and R. Stinson (eds), John Wiley, 1992, pp. 241–324.
Karzel, H.: Ebene Inzidenzgruppen,Arch. Math. 40 (1964), 10–17.
Kantor, W.: Primitive permutation groups of odd order and an application to finite projective planes,J. Algebra 106 (1987), 15–45.
Lander, E.: Symmetric designs: An algebraic approach,London Math. Soc. Lecture Note 74 (1983).
Ljunggren, W.: Einige Bemerkungen über die Darstellung ganzer Zahlen durch binäre kubische Formen mit positiver Diskriminante,Acta Math. 75 (1942), 1–21.
Ostrom, T. G.: Concerning difference sets,Canad. J. Math. 5 (1953), 421–424.
Ott, U.: Endliche Zyklische Ebenen,Math. Z. 53 (1975), 195–215.
Singer, J.: A theorem in finite projective geometry and some applications to number theory,Trans. Amer. Math. Soc. 43 (1938), 377–385.
Thompson, J. G.: Finite groups with fixed-point-free automorphisms of prime order,Proc. Nat. Acad. Sci. U.S.A. 45 (1956), 578–581.
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