Primitive bicirculant association schemes and a generalization of Wielandt’s theorem
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- by I. Kovács, D. Marušič and M. Muzychuk PDF
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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 $e \geq 1$, 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 $p^e-1$.References
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Additional Information
- I. Kovács
- Affiliation: FAMNIT, University of Primorska, Glagoljaška 8, 6000 Koper, Slovenia
- Email: kovacs@pef.upr.si
- D. Marušič
- 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
- MR Author ID: 249196
- Email: muzy@netanya.ac.il
- Received by editor(s): May 22, 2007
- Received by editor(s) in revised form: June 26, 2008
- Published electronically: January 7, 2010
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 362 (2010), 3203-3221
- MSC (2000): Primary 05E30
- DOI: https://doi.org/10.1090/S0002-9947-10-04864-6
- MathSciNet review: 2592953