Multiplier groups of planar difference sets and a theorem of Kantor

Ho, Chat Yin; Pott, Alexander

doi:10.1090/S0002-9939-1990-1017847-7

Multiplier groups of planar difference sets and a theorem of Kantor
HTML articles powered by AMS MathViewer

by Chat Yin Ho and Alexander Pott PDF

Proc. Amer. Math. Soc. 109 (1990), 803-808 Request permission

Abstract:

A recent result of W. Kantor followed by a work of W. Feit has rekindled interest in the longstanding conjecture of finite cyclic planes. In this paper we prove that the order of the multiplier group equals the odd part of the order of the automorphism group of a Singer group if and only if the order of the plane is 2, 3, or 8. This yields another proof for Feit’s result mentioned above.

References

Leonard D. Baumert, Cyclic difference sets, Lecture Notes in Mathematics, Vol. 182, Springer-Verlag, Berlin-New York, 1971. MR 0282863
Walter Feit, Finite projective planes and a question about primes, Proc. Amer. Math. Soc. 108 (1990), no. 2, 561–564. MR 1002157, DOI 10.1090/S0002-9939-1990-1002157-4
Daniel Gorenstein, Finite groups, Harper & Row, Publishers, New York-London, 1968. MR 0231903
Chat Y. Ho, On multiplier groups of finite cyclic planes, J. Algebra 122 (1989), no. 1, 250–259. MR 994947, DOI 10.1016/0021-8693(89)90249-4
Daniel R. Hughes and Fred C. Piper, Projective planes, Graduate Texts in Mathematics, Vol. 6, Springer-Verlag, New York-Berlin, 1973. MR 0333959
Dieter Jungnickel and Klaus Vedder, On the geometry of planar difference sets, European J. Combin. 5 (1984), no. 2, 143–148. MR 753004, DOI 10.1016/S0195-6698(84)80028-1
William M. Kantor, Primitive permutation groups of odd degree, and an application to finite projective planes, J. Algebra 106 (1987), no. 1, 15–45. MR 878466, DOI 10.1016/0021-8693(87)90019-6
Udo Ott, Endliche zyklische Ebenen, Math. Z. 144 (1975), no. 3, 195–215 (German). MR 380619, DOI 10.1007/BF01214135