Cayley graphs on abelian groups

Abstract

Let A be an abelian group and let ι be the automorphism of A defined by: ι: a ↦ a−1. A Cayley graph Γ = Cay(A,S) is said to have an automorphism group as small as possible if Aut(Γ)=A⋊<ι>. In this paper, we show that almost all Cayley graphs on abelian groups have automorphism group as small as possible, proving a conjecture of Babai and Godsil.

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References

  1. [1]

    L. Babai: On a conjecture of M. E. Watkins on graphical regular representations offinite groups, Compositio Math. 37 (1978), 291–296.

    MathSciNet  MATH  Google Scholar 

  2. [2]

    L. Babai: Finite digraphs with given regular automorphism groups, Period. Math.Hungar. 11 (1980), 257–270.

    MathSciNet  Article  MATH  Google Scholar 

  3. [3]

    L. Babai and C. D. Godsil: On the automorphism groups of almost all Cayleygraphs, European J. Combin. 3 (1982), 9–15.

    MathSciNet  Article  MATH  Google Scholar 

  4. [4]

    S. Bhoumik, E. Dobson and J. Morris: On The Automorphism Groups of AlmostAll Circulant Graphs and Digraphs, Ars Math. Contemp. 7 (2014), 487–506.

    MathSciNet  MATH  Google Scholar 

  5. [5]

    C. Casolo, E. Jabara and P Spiga: On the Fitting height of factorised solublegroups, J. Group Theory 17 (2014), 911–924.

    MathSciNet  Article  MATH  Google Scholar 

  6. [6]

    J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson: Atlas of Finite Groups, Clarendon Press, Oxford, 1985.

    Google Scholar 

  7. [7]

    E. Dobson: Asymptotic automorphism groups of Cayley digraphs and graphs ofabelian groups of prime-power order, Ars Math. Contemp. 3 (2010), 200–213.

    MathSciNet  Google Scholar 

  8. [8]

    S. A. Evdokimov and I. N. Ponomarenko: Characterization of cyclotomic schemesand normal Schur rings over a cyclic group, Algebra i Analiz 14 (2002), 11–55.

    MATH  Google Scholar 

  9. [9]

    C. D. Godsil: GRRs for nonsolvable groups, Algebraic methods in graph theory, Vol. I, II (Szeged, 1978), 221–239, Colloq. Math. Soc. János Bolyai, Amsterdam-New York, 1981.

    Google Scholar 

  10. [10]

    C. D. Godsil: On the full automorphism group of a graph, Combinatorica 1 (1981), 243–256.

    MathSciNet  Article  MATH  Google Scholar 

  11. [11]

    D. Hetzel: über regulare graphische Darstellungen von auosbaren Gruppen, Technische Universität, Berlin, 1976. (Diplomarbeit)

    Google Scholar 

  12. [12]

    W. Imrich: Graphical regular representations of groups of odd order, Combinatorics(Proc. Colloq., Keszthely, 1976), Bolyai-North-Holland, 1978, 611–621.

    Google Scholar 

  13. [13]

    M. Isaacs: Finite Group Theory, Graduate Studies in Mathematics 92, (2008).

    Google Scholar 

  14. [14]

    E. Jabara and P. Spiga: Abelian Carter subgroups in finite permutation groups,Arch. Math. 101 (2013), 301–307.

    MathSciNet  Google Scholar 

  15. [15]

    H. Kurzweil and B. Stellmacher: The Theory of Finite Groups, An Introduction, Universitext, Springer 2004.

    Google Scholar 

  16. [16]

    K. H. Leung and S. H. Man: On Schur rings over cyclic groups. II, J. Algebra 183 (1996), 273–285.

    MathSciNet  Article  MATH  Google Scholar 

  17. [17]

    K. H. Leung and S. H. Man: On Schur rings over cyclic groups, Israel J. Math. 106 (1998), 251–267.

    MathSciNet  Article  MATH  Google Scholar 

  18. [18]

    C. H. Li: Permutation groups with a cyclic regular subgroup and arc-transitive circulants, J. Algebraic Combin. 21 (2005), 131–136.

    MathSciNet  Article  MATH  Google Scholar 

  19. [19]

    C. H. Li and H. Zhang: The finite primitive groups with soluble stabilizers, and theedge-primitive s-arc transitive graphs, Proc. London Math. Soc. 103 (2011), 441–472.

    Article  MATH  Google Scholar 

  20. [20]

    M. W. Liebeck, C. E. Praeger and J. Saxl: On the O'Nan-Scott theorem forfinite primitive permutation groups, J. Austral. Math. Soc. Ser. A 44 (1988), 389–396.

    MathSciNet  Article  MATH  Google Scholar 

  21. [21]

    M. W. Liebeck, C. E. Praeger and J. Saxl: The maximal factorizations of thefinite simple groups and their automorphism groups, Mem. Amer. Math. Soc. 86 (1990).

  22. [22]

    J. Morris, P. Spiga and G Verret: Automorphisms of Cayley graphs on generaliseddicyclic groups, European J. Combin. 43 (2015), 68–81.

    MathSciNet  Article  MATH  Google Scholar 

  23. [23]

    L. A. Nowitz: On the non-existence of graphs with transitive generalized dicyclicgroups, J. Combinatorial Theory, 4 (1968), 49–51.

    MathSciNet  Article  MATH  Google Scholar 

  24. [24]

    L. A. Nowitz and M. E. Watkins: Graphical regular representations of non-abeliangroups I-II, Canad. J. Math. 24 (1972), 993–1008 and 1009–1018.

    MathSciNet  Article  MATH  Google Scholar 

  25. [25]

    P. PotoČnik, P. Spiga and G. Verret: Asymptotic enumeration of vertex transitive graphs of fixed valency, arXiv:1210.5736 [math.CO].

  26. [26]

    M. Suzuki: Group Theory I, Springer-Verlag, 1982.

    Google Scholar 

  27. [27]

    M.Y. Xu: Automorphism groups and isomorphisms of Cayley digraphs, DiscreteMath. 182 (1998), 309–319.

    MathSciNet  MATH  Google Scholar 

  28. [28]

    T. Yoshida: Character theoretic transfer, J. Algebra 52 (1978), 1–38.

    MathSciNet  Article  MATH  Google Scholar 

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Correspondence to Gabriel Verret.

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The third author is supported by UWA as part of the Australian Research Council grant DE130101001.

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Dobson, E., Spiga, P. & Verret, G. Cayley graphs on abelian groups. Combinatorica 36, 371–393 (2016). https://doi.org/10.1007/s00493-015-3136-5

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Mathematics Subject Classification (2010)

  • 20B25
  • 05E18