Skip to main content

On the cayley index of a group

Invited Addresses

  • 561 Accesses

Part of the Lecture Notes in Mathematics book series (LNM,volume 686)

Abstract

The set of left multiplications by the elements of a group G form a subgroup LG of the automorphism group A(X) for any Cayley graph X of G. The Cayley index c(G) of G is the minimum of the index [A(X) : LG] taken over all Cayley graphs X of G. The present article is expository, reporting what is known to date about Cayley indices of groups and where some of the results may be found. The general pattern thus far indicates that, with finitely many exceptions, c(G) is determined by whether G is (1) abelian, (2) generalized dicyclic, (3) solvable and finite but neither abelian nor generalized dicyclic, or (4) none of the above.

Keywords

  • Abelian Group
  • Automorphism Group
  • Cayley Graph
  • Free Product
  • Solvable Group

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   34.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   46.00
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. L. Babai, "On a conjecture of M.E. Watkins", (to appear).

    Google Scholar 

  2. E. Bannai, "Graphical regular representations of non-solvable groups", (to appear).

    Google Scholar 

  3. C.-Y. Chao, "On a theorem of Sabidussi", Proc. Amer. Math. Soc. 15 (1964), 291–294.

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. D. Hetzel, "Über reguläre graphische Darstellung von auflösbaren Gruppen", Diplomarbeit, Technische Universität Berlin, (1977), 371 pp.

    Google Scholar 

  5. W. Imrich, "Groups with transitive Abelian automorphism groups", in Combinatorial Theory and Its Applications, Coll. Soc. Janós, Bolyai 4, Balatonfűred, Hungary, (1969), 651–656.

    Google Scholar 

  6. W. Imrich, "On graphs with regular groups", J. Combinatorial Theory (B) 19 (1975), 174–180.

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. W. Imrich and M.E. Watkins, "On graphical regular representations of cyclic extensions of groups", Pacific J. Math. 54 (1974), 149–165.

    MathSciNet  MATH  Google Scholar 

  8. W. Imrich and M.E. Watkins, "On automorphism groups of Cayley graphs", Per. Math. Hung. (to appear).

    Google Scholar 

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

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. L.A. Nowitz and M.E. Watkins, "Graphical regular representations of non-abelian groups, I", Canad. J. Math. 24 (1972), 993–1008.

    CrossRef  MathSciNet  MATH  Google Scholar 

  11. L.A. Nowitz and M.E. Watkins, "Graphical regular representations of non-abelian groups, II", Canad. J. Math. 24 (1972), 1009–1018.

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. G. Sabidussi, "On a class of fixed-point free graphs", Proc. Amer. Math. Soc. 9 (1958), 800–804.

    CrossRef  MathSciNet  MATH  Google Scholar 

  13. G. Sabidussi, "Vertex-transitive graphs", Monatsh. Math. 68 (1964), 426–438.

    CrossRef  MathSciNet  MATH  Google Scholar 

  14. M.E. Watkins, "On the action of non-abelian groups on graphs", J. Combinatorial Theory B 1 (1971), 95–104.

    CrossRef  MathSciNet  MATH  Google Scholar 

  15. M.E. Watkins, "On graphical regular representations of Cn × Q", in Graph Theory and Applications (Ed. Y. Alavi, D.R. Lick, A.T. White), Springer-Verlag, Berlin, (1972), 305–311.

    CrossRef  Google Scholar 

  16. M.E. Watkins, "Graphical regular representations of alternating, symmetric, and miscellaneous small groups", Aequat. Math. 11 (1974), 40–50.

    CrossRef  MathSciNet  MATH  Google Scholar 

  17. M.E. Watkins, "Graphical regular representations of free products of groups", J. Combinatorial Theory (B) 21 (1976), 47–56.

    CrossRef  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1978 Springer-Verlag

About this paper

Cite this paper

Watkins, M.E. (1978). On the cayley index of a group. In: Holton, D.A., Seberry, J. (eds) Combinatorial Mathematics. Lecture Notes in Mathematics, vol 686. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0062521

Download citation

  • DOI: https://doi.org/10.1007/BFb0062521

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-08953-7

  • Online ISBN: 978-3-540-35702-5

  • eBook Packages: Springer Book Archive