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Semiregular elements in cubic vertex-transitive graphs and the restricted Burnside problem

Published online by Cambridge University Press:  15 April 2014

PABLO SPIGA
PABLO SPIGA*
Affiliation:
Dipartimento di Matematica Pura e Applicata, University of Milano-Bicocca, Via Cozzi 55, 20126 Milano, Italy. e-mail: [email protected]
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Abstract

In this paper, we prove that the maximal order of a semiregular element in the automorphism group of a cubic vertex-transitive graph Γ does not tend to infinity as the number of vertices of Γ tends to infinity. This gives a solution (in the negative) to a conjecture of Peter Cameron, John Sheehan and the author [4, conjecture 2].

However, with an application of the positive solution of the restricted Burnside problem, we show that this conjecture holds true when Γ is either a Cayley graph or an arc-transitive graph.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 157 , Issue 1 , July 2014 , pp. 45 - 61
Copyright
Copyright © Cambridge Philosophical Society 2014 

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References

REFERENCES

[1]Bosma, W., Cannon, J. and Playoust, C.The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), 235–265.Google Scholar
[2]Cameron, P. J.Permutation groups. London Mathematical Society Student Texts 45 (Cambridge University Press, 1999).CrossRefGoogle Scholar
[3]Cameron, P., Giudici, M., Jones, G. A., Kantor, W., Klin, M., Marušič, D. and Nowitz, L. A.. Transitive permutation groups without semiregular subgroups. J. London Math. Soc. (2) 66 (2002), 325–333.Google Scholar
[4]Cameron, P., Sheehan, J. and Spiga, P.Semiregular automorphisms of vertex-transitive cubic graphs. European J. Combin. 27 (2006), 924–930.Google Scholar
[5]Carter, R. W.Simple Groups of Lie Type. (Wiley Classics Library, John Wiley & Sons, Inc., New York, 1989).Google Scholar
[6]Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A. and Wilson, R. A.. Atlas of Finite Groups. (Claredon Press, Oxford, 1985).Google Scholar
[7]Dobson, E., Malnič, A., Marušič, D. and Nowitz, L. A.. Semiregular automorphisms of vertex-transitive graphs of certain valencies. J. Combin. Theory Ser. B 97 (2007), 371–380.Google Scholar
[8]Giudici, M. and Xu, J.All vertex-transitive locally-quasiprimitive graphs have a semiregular automorphism. J. Algebraic Combin. 25 (2007), 217–232.Google Scholar
[9]Gorenstein, D., Lyons, R. and Solomon, R.The Classification of the Finite Simple Groups, Number 3. Mathematical Surveys and Monographs, Vol. 40. (American Mathematical Society, 1994).CrossRefGoogle Scholar
[10]Hall, P. and Higman, G.On the p-length of p-soluble groups and reduction theorems for Burnside's problem. Proc. London Math. Soc. 6 (1956), 1–42.Google Scholar
[11]Havas, G., Newman, M. F., Neimeyer, A. C. and Sims, C. C.. Groups with exponent six. Comm. Algebra 27 (1999), 3619–3638.CrossRefGoogle Scholar
[12]Jordan, D.Eine Symmetrieeigenschaft von Graphen, Graphentheorie und ihre Anwendungen. Dresdner Reihe Forsch. 9 (Dresden, 1988).Google Scholar
[13]Klin, M. On transitive permutation groups without semi-regular subgroups, ICM 1998: International Congress of Mathematicians. (Berlin, 18–27 August 1998). Abstracts of short communications and poster sessions. (1998), 279.Google Scholar
[14]Kutnar, K. and Šparl, P.. Distance-transitive graphs admit semiregular automorphisms. European J. Combin. 31 (2010), 25–28.Google Scholar
[15]Marušič, D.On vertex symmetric digraphs. Discrete Math. 36 (1981), 69–81.CrossRefGoogle Scholar
[16]Marušič, D. and Scapellato, R.Permutation groups, vertex-transitive digraphs and semiregular automorphisms. European J. Combin. 19 (1998) 707–712Google Scholar
[17]McKay, B. and Praeger, C. E.Vertex-transitive graphs which are not Cayley graphs. I. J. Austral. Math. Soc. Ser. A 56 (1994), 53–63.CrossRefGoogle Scholar
[18]Morris, J., Spiga, P. and Verret, G. Semiregular automorphisms of cubic vertex-transitive graphs. Submitted.Google Scholar
[19]Potočnik, P., Spiga, P. and Verret, G.Cubic vertex-transitive graphs on up to 1280 vertices. J. Symbolic Comput. (2013), 465–477. http://dx.doi.org/10.1016/j.jsc.2012.09.00.Google Scholar
[20]Potocnik, P., Spiga, P. and Verret, G. Asymptotic enumeration of vertex-transitive graphs of fixed valency. Submitted.Google Scholar
[21]Praeger, C. E., Spiga, P. and Verret, G.Bounding the size of a vertex-stabiliser in a finite vertex-transitive graph. J. Combin. Theory Ser. B 102 (2012), 797–819.CrossRefGoogle Scholar
[22]Robinson, D. J. S.A course in the theory of groups. Graduate Texts in Mathematics 20 (Springer-Verlag, 1982).Google Scholar
[23]Tutte, W. T.A family of cubical graphs. Proc. Camb. Phil. Soc. 43 (1947), 459–474.CrossRefGoogle Scholar
[24]Vaughan-Lee, M.The Restricted Burnside Problem, Second Edition, London Mathematical Society Monographs N.S. 8 (Oxford Science Publications, 2003).Google Scholar
[25]Zelmanov, E. I.The Solution of the Restricted Burnside problem for groups of odd exponent. Izv. Math. USSR 36 (1991), 4–60.Google Scholar
[26]Zelmanov, E. I.The solution of the restricted Burnside problem for 2-groups. Mat. Sb. 182 (1991), 568–592.Google Scholar
[27]Zsigmondy, K.Zur theorie der Potenzreste. Monatsh. Math. Phys. 3 (1892), 265–284.CrossRefGoogle Scholar