Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-10-04864-6
Published electronically: January 7, 2010
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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$.


References [Enhancements On Off] (What's this?)

  • 1. E. Bannai and T. Ito, Algebraic combinatorics I: Association schemes, W. A. Benjamin, Menlo Park, CA, 1984. MR 882540 (87m:05001)
  • 2. R. C. Bose and T. Shimamoto, Classification and analysis of partially balanced incomplete block designs with two associate classes, J. Amer. Statist. Assoc. 47 (1952), 151-184. MR 0048772 (14:67b)
  • 3. A. E. Brouwer, A. E. Cohen, and A. Neumaier, Distance-regular graphs, Springer-Verlag, 1989. MR 1002568 (90e:05001)
  • 4. C. W. Curtis and I. Reiner, ``Representation theory of finite groups and associative algebras'', John Wiley & Sons, New York, London, 1962. MR 0144979 (26:2519)
  • 5. I. A. Faradžev, M. H. Klin and M. E. Muzychuk, Cellular rings and automorphism groups of graphs In: Investigations on Algebraic Theory of Combinatorial Objects, Mathematics and its Applications (Soviet Series), v. 84, I. A. Faraďev, A. A. Ivanov, M. H. Klin, A. J. Woldar (Eds.), Kluwer Acad. Publ., 1994. MR 1273366 (95a:05049)
  • 6. R. Guralnick, Private communication, 2008.
  • 7. N. Ito, On transitive simple permutation groups of degree $ 2p$, Math. Z. 178 (1962), 453-468. MR 0140564 (25:3982)
  • 8. N. Ito, On uniprimitive groups of degree $ 2p$, Math. Z. 102 (1967), 238-244. MR 0219602 (36:2681)
  • 9. N. Ito and W. Tomoyuki, A note on transitive permutation groups of degree $ 2p$, Tensor (N.S.) 26 (1972), 105-106. MR 0330271 (48:8608)
  • 10. I. Kovács, A. Malnič, D. Marušič and Š. Miklavič, Transitive group actions: (Im)primitivity and semiregular subgroups, submitted paper.
  • 11. W. Knapp, On Burnside's Method, J. Algebra 175 (1995), 644-660. MR 1339661 (96i:20002)
  • 12. K. H. Leung and S. L. Ma, Partial difference triples, J. Algebr. Combin. 2 (1993), 397-409. MR 1241508 (94h:05099)
  • 13. M.W. Liebeck and J. Saxl, The finite primitive permutation groups of rank three, Bull. London Math. Soc. 18 (1986), 165-172. MR 818821 (87i:20007)
  • 14. A. Malnič, D. Marušič and P. Šparl, On strongly regular bicirculants, Europ. J. Combin. 28 (2007), 891-900. MR 2300769 (2007m:05121)
  • 15. D. Marušič, Strongly regular bicirculants and tricirculants, Ars Combin. 25C (1988), 11-15. MR 943371 (89e:05105)
  • 16. P. Müller, Permutation Groups with a Cyclic Two-Orbits Subgroup and Monodromy Groups of Siegel Functions, http://arxiv.org/PS-cache/math/pdf/0110/0110060v1.pdf
  • 17. M. J. de Resmini and D. Jungnickel, Strongly regular semi-Cayley graphs, J. Algebr. Combin. 1 (1992), 171-195. MR 1226350 (94d:05150)
  • 18. L. L. Scott, On primitive permutation groups of degree $ 2p$, Math. Z. 126 (1972), 227-229. MR 0346034 (49:10760)
  • 19. L. L. Scott, Estimates in permutation groups, Geom. Dedicata 5 (1976), 219-227. MR 0424911 (54:12869)
  • 20. W. R. Scott, Group Theory, Dover Publications, New York, 1987. MR 896269 (88d:20001)
  • 21. H. Wielandt, Zur Theorie der einfach transitiven Permutationsgruppen II, Math. Z. 52 (1949), 384-393. MR 0033817 (11:495a)
  • 22. H. Wielandt, ``Finite Permutation Groups'', Academic Press, New York, 1964. MR 0183775 (32:1252)
  • 23. H. Wielandt, Primitive Permutationsgruppen vom Grad $ 2p$, Math. Z. 63 (1956), 478-485. MR 0075200 (17:708c)
  • 24. P. H. Zieschang, ``An algebraic approach to association schemes'', Lecture Notes in Math., Vol. 1628, Springer-Verlag, New York/Berlin, 1996. MR 1439253 (98h:05185)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 05E30

Retrieve articles in all journals with MSC (2000): 05E30


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: https://doi.org/10.1090/S0002-9947-10-04864-6
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.

American Mathematical Society