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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Primitive bicirculant association schemes and a generalization of Wielandt's theorem


Authors: I. Kovács, D. Marusic and M. Muzychuk
Journal: Trans. Amer. Math. Soc. 362 (2010), 3203-3221
MSC (2000): Primary 05E30
Published electronically: January 7, 2010
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Abstract: Bannai and Ito defined association scheme theory as doing ``group theory without groups'', thus raising a basic question as to which results about permutation groups are, in fact, results about association schemes. By considering transitive permutation groups in a wider setting of association schemes, it is shown in this paper that one such result is the classical theorem of Wielandt about primitive permutation groups of degree $ 2p$, $ p$ a prime, being of rank at most $ 3$ (see Math. Z. 63 (1956), 478-485). More precisely, it is proved here that if $ \mathfrak{X}$ is a primitive bicirculant association scheme of order $ 2p^e$, $ p>2$ is a prime, then $ \mathfrak{X}$ is of class at most $ 2$, and if it is of class exactly $ 2$, then $ 2p^e=(2s+1)^2+1$ for some natural number $ s$, with the valencies of $ \mathfrak{X}$ being $ 1$, $ s(2s+1)$, $ (s+1)(2s+1)$, and the multiplicities of $ \mathfrak{X}$ being $ 1$, $ p^e$, $ p^e-1$. Consequently, translated into permutation group theory language, a primitive permutation group $ G$ of degree $ 2p^e$, $ p$ a prime and $ e \geq1$, containing a cyclic subgroup with two orbits of size $ p^e$, is either doubly transitive or of rank $ 3$, in which case $ 2p^e=(2s+1)^2+1$ for some natural number $ s$, the sizes of suborbits of $ G$ are $ 1$, $ s(2s+1)$ and $ (s+1)(2s+1)$, and the degrees of the irreducible constituents of $ G$ are $ 1$, $ p^e$ and $ p^e-1$.


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Additional Information

I. Kovács
Affiliation: FAMNIT, University of Primorska, Glagoljaška 8, 6000 Koper, Slovenia
Email: kovacs@pef.upr.si

D. Marusic
Affiliation: IMFM, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia – and – FAMNIT, University of Primorska, Glagoljaška 8, 6000 Koper, Slovenia
Email: dragan.marusic@guest.arnes.si

M. Muzychuk
Affiliation: Department of Computer Science and Mathematics, Netanya Academic College, 1 University St., 42365 Netanya, Israel
Email: muzy@netanya.ac.il

DOI: http://dx.doi.org/10.1090/S0002-9947-10-04864-6
PII: S 0002-9947(10)04864-6
Received by editor(s): May 22, 2007
Received by editor(s) in revised form: June 26, 2008
Published electronically: January 7, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.