Skip to main content

Some Unexpected Consequences of Symmetry Computations

  • 595 Accesses

Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS,volume 159)

Abstract

This paper gives some instances of experimental computations involving the action of groups on graphs and maps with a high degree of symmetry, that have led to unexpected theoretical discoveries. These include new presentations for 3-dimensional special linear groups, a closed-form definition for the binary reflected Gray codes, a new theorem on groups expressible as a product of an abelian group and a cyclic group, and some revealing observations about the genus spectrum of particular classes of regular maps on surfaces.

Keywords

  • Normal Subgroup
  • Automorphism Group
  • Cayley Graph
  • Gray Code
  • Genus Spectrum

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 (USA)
  • DOI: 10.1007/978-3-319-30451-9_3
  • Chapter length: 9 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   169.00
Price excludes VAT (USA)
  • ISBN: 978-3-319-30451-9
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   219.99
Price excludes VAT (USA)
Hardcover Book
USD   219.99
Price excludes VAT (USA)
Fig. 1
Fig. 2

References

  1. K. Appel and W. Haken, Every planar map is four colorable. Part I. Discharging, Illinois J. Math. 21 (1977), 429–490.

    Google Scholar 

  2. K. Appel, W. Haken and J. Koch, Every planar map is four colorable. Part II. Reducibility, Illinois J. Math. 21 (1977), 491–567.

    Google Scholar 

  3. A.B. D’Azevedo, R. Nedela and J. Širáň, Classification of regular maps of negative prime Euler characteristic, Trans. Amer. Math. Soc. 357 (2005), 4175–4190.

    Google Scholar 

  4. N. Biggs, Presentations for cubic graphs, in Computational Group Theory (ed. M. Atkinson), Academic Press, 1984, pp. 57–63.

    Google Scholar 

  5. W. Bosma, J. Cannon and C. Playoust, The MAGMA Algebra System I: The User Language, J. Symbolic Computation 24 (1997), 235–265.

    Google Scholar 

  6. H.R. Brahana, Regular maps and their groups, Amer. J. Math. 49 (1927), 268–284.

    Google Scholar 

  7. M.D.E. Conder, A surprising isomorphism, J.Algebra 129 (1990) 494–501.

    Google Scholar 

  8. M.D.E. Conder, Explicit definition of the binary reflected Gray codes, Discrete Math. 195 (1999), 245–249.

    Google Scholar 

  9. M.D.E. Conder, Regular maps and hypermaps of Euler characteristic 1 to 200, J. Combinatorial Theory, Ser. B 99 (2009), 455–459.

    Google Scholar 

  10. M.D.E. Conder and I.M. Isaacs, Derived subgroups of products of an abelian and a cyclic subgroup, J. London Math. Society 69 (2004), 333–348.

    Google Scholar 

  11. M.D.E. Conder, R. Jajcay and T.W. Tucker, Regular Cayley maps for finite abelian groups,J. Algebraic Combinatorics 25 (2007), 259–283.

    Google Scholar 

  12. M.D.E. Conder and P.J. Lorimer, Automorphism groups of symmetric graphs of valency 3,J. Combinatorial Theory, Ser. B 47 (1989), 60–72.

    Google Scholar 

  13. M.D.E. Conder, E.F. Robertson and P.R. Williams, Presentations for 3-dimensional special linear groups over integer rings, Proc. Amer. Math. Soc. 115 (1992), 19–26.

    Google Scholar 

  14. M.D.E. Conder, J. Širáň and T.W. Tucker, The genera, reflexibility and simplicity of regular maps, J. European Math. Soc. 12 (2010), 343–364.

    Google Scholar 

  15. M.D.E. Conder and T.W. Tucker, Regular Cayley maps for cyclic groups, Trans. Amer. Math. Soc. 366 (2014), 3585–3609.

    Google Scholar 

  16. M.D.E. Conder and C.G. Walker, Vertex-transitive graphs with arbitrarily large vertexstabilizers, J. Algebraic Combinatorics 8 (1998), 29–38.

    Google Scholar 

  17. H.S.M. Coxeter and W.O.J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer (1980).

    Google Scholar 

  18. D. Ž. Djoković and G.L. Miller, Regular groups of automorphisms of cubic graphs, J. Combinatorial Theory Ser. B 29 (1980), 195–230.

    Google Scholar 

  19. D. Firth, An algorithm to find normal subgroups of a finitely presented group up to a given index, PhD Thesis, University of Warwick, 2005.

    Google Scholar 

  20. N. Itô, Über das Produkt von zwei abelschen Gruppen. Math. Zeitschrift 62 (1955), 400–401.

    Google Scholar 

  21. G.A. Jones and D. Singerman, Belyĭ functions, hypermaps, and Galois groups, Bull. London Math. Soc. 28 (1996), 561–590.

    Google Scholar 

  22. N. Robertson, D. Sanders, P. Seymour and R. Thomas, The four-colour theorem, J. Combin. Theory Ser. B 70 (1997), 2–44.

    Google Scholar 

  23. W.T. Tutte, On the symmetry of cubic graphs, Canad. J. Math. 11 (1959), 621–624.

    Google Scholar 

Download references

Acknowledgments

The author is grateful for support for this work from the Marsden Fund and a James Cook Fellowship of the Royal Society of New Zealand, and acknowledges the considerable help of Magma [5] in showing what was possible.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Marston D. E. Conder .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Conder, M.D.E. (2016). Some Unexpected Consequences of Symmetry Computations. In: Širáň, J., Jajcay, R. (eds) Symmetries in Graphs, Maps, and Polytopes. SIGMAP 2014. Springer Proceedings in Mathematics & Statistics, vol 159. Springer, Cham. https://doi.org/10.1007/978-3-319-30451-9_3

Download citation