On schemes originated from Ferrero pairs

Document Type: Research Paper

Authors

1 Department of‎ ‎Mathematics‎, K‎. ‎N‎. ‎Toosi University of Technology‎, ‎‎

2 K. N. Toosi university of Technology University, Tehran-Iran.

Abstract

‎‎The Frobenius complement of a given Frobenius group acts on its kernel‎. ‎The scheme which is arisen from the orbitals of this action is called Ferrero pair scheme‎. ‎In this paper‎, ‎we show that the fibers of a Ferrero pair scheme consist of exactly one singleton fiber and every two fibers with more than one point have the same cardinality‎. ‎Moreover‎, ‎it is shown that the restriction of a Ferrero pair scheme on each fiber is isomorphic to a regular scheme‎. ‎Finally‎, ‎we prove that for any prime $p$‎, ‎there exists a Ferrero pair $p$-scheme‎, ‎and if $p> 2$‎, ‎then the Ferrero pair $p$-schemes of the same rank are all isomorphic‎.

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K. I. Beidar, W. F. Ke and H. Kiechle (2007). Automorphisms of certain design groups II. J. Algebra. 313 (2), 672-686
R. Brown (2001). Frobenius groups and classical maximal orders. Mem. Amer. Math. Soc.. 151 (717), 0-110
J. D. Dixon and B. Mortimer (1996). Permutation Groups. Graduate Texts in Mathematics, Springer-Verlag, New York. 163
S. A. Evdokimov and I. N. Ponomarenko (2003). Characterization of cyclotomic schemes and normal Schur rings over a cyclic group. Algebra i Analiz, 14 no. 2 (2002) 11-55, translation in St. Petersburg Math. J.. 14 (2), 189-221
S. Evdokimov and I. Ponomarenko (2009). Permutation group approach to association schemes. European J. Combin.. 30, 1456-1476
B. Huppert (1998). Character theory of finite groups. de Gruyter Expositions in Mathematics, 25 Walter de Gruyter $&$ Co., Berlin. 25
D. S. Passman (1968). Permutation Groups. W. A. Benjamin, Inc., New York-Amsterdam.
I. N. Ponomarenko and A. Rahnamai Barghi (2007). On the structure of $p$-schemes. (Russian) Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 344} (2007), translation in J. Math. Sci. (N. Y.). 147 (6), 7227-7233
J. A. Wolf (1967). Spaces of Constant Curvature. McGraw-Hill Book Co., New York-London-Sydney.
P. H. Zieschang (2005). Theory of Association Schemes. Springer Monographs in Mathematic, Springer-Verlag, Berlin.