Hostname: page-component-7c8c6479df-8mjnm Total loading time: 0 Render date: 2024-03-28T10:57:50.137Z Has data issue: false hasContentIssue false

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

Published online by Cambridge University Press:  15 April 2014

PABLO SPIGA*
Affiliation:
Dipartimento di Matematica Pura e Applicata, University of Milano-Bicocca, Via Cozzi 55, 20126 Milano, Italy. e-mail: pablo.spiga@unimib.it

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
Copyright
Copyright © Cambridge Philosophical Society 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Bosma, W., Cannon, J. and Playoust, C.The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), 235265.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), 325333.Google Scholar
[4]Cameron, P., Sheehan, J. and Spiga, P.Semiregular automorphisms of vertex-transitive cubic graphs. European J. Combin. 27 (2006), 924930.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), 371380.Google Scholar
[8]Giudici, M. and Xu, J.All vertex-transitive locally-quasiprimitive graphs have a semiregular automorphism. J. Algebraic Combin. 25 (2007), 217232.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), 142.Google Scholar
[11]Havas, G., Newman, M. F., Neimeyer, A. C. and Sims, C. C.. Groups with exponent six. Comm. Algebra 27 (1999), 36193638.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), 2528.Google Scholar
[15]Marušič, D.On vertex symmetric digraphs. Discrete Math. 36 (1981), 6981.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), 5363.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), 465477. 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), 797819.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), 459474.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), 460.Google Scholar
[26]Zelmanov, E. I.The solution of the restricted Burnside problem for 2-groups. Mat. Sb. 182 (1991), 568592.Google Scholar
[27]Zsigmondy, K.Zur theorie der Potenzreste. Monatsh. Math. Phys. 3 (1892), 265284.CrossRefGoogle Scholar