Skip to main content Accessibility help
×
Home

On isomorphisms of connected Cayley graphs, III

  • Cai Heng Li (a1)

Abstract

For a finite group G and a subset S of G which does not contain the identity of G, we use Cay(G, S) to denote the Cayley graph of G with respect to S. For a positive integer m, the group G is called a (connected) m-DCI-group if for any (connected) Cayley graphs Cay(G, S) and Cay(G, T) of out-valency at most m, Sσ = T for some σ ∈ Aut(G) whenever Cay(G, S) ≅ Cay(G, T). Let p(G) be the smallest prime divisor of |G|. It was previously shown that each finite group G is a connected m-DCI-group for mp(G) − 1 but this is not necessarily true for m = p(G). This leads to a natural question: which groups G are connected p(G)-DCI-groups? Here we conjecture that the answer of this question is positive for finite simple groups, that is, finite simple groups are all connected 2-DCI-groups. We verify this conjecture for the linear groups PSL(2, q). Then we prove that a nonabelian simple group G is a 2-DCI-group if and only if G = A5.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      On isomorphisms of connected Cayley graphs, III
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      On isomorphisms of connected Cayley graphs, III
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      On isomorphisms of connected Cayley graphs, III
      Available formats
      ×

Copyright

References

Hide All
[1]Alspach, B., ‘Isomorphisms of Cayley graphs on Abelian groups’, in Graph Symmetry: Algebraic Methods and Applications 497, NATO Adv. Sc. Inst. Ser. C (Kluwer Acad. Publ., Dordrecht, 1997), pp. 123.
[2]Alspach, B. and Parsons, T.D., ‘Isomorphisms of circulant graphs and digraphs’, Discrete Math. 25 (1979), 97108.
[3]Babai, L., Isomorphism problem for a class of point-symmetric structures, Acta Math. Acad. Sci. Hungar. 29 (1977), 329336.
[4]Biggs, N., Algebraic graph theory (Cambridge University Press, New York, 1992).
[5]Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A. and Wilson, R.A., Atlas of finite groups (Clarendon Press, Oxford, 1985).
[6]Delorme, C., Favaron, O. and Maheo, M., ‘Isomorphisms of Cayley multigraphs of degree 4 on finite Abelian groups’, European J. Combin. 13 (1992), 5961.
[7]Godsil, C.D., ‘On the full automorphism group of a graph’, Combinatorica 1 (1981), 243256.
[8]Guralnick, R., ‘Subgroups of prime power index in a simple group’, J. Algebra 81 (1983), 304311.
[9]Huppert, B. and Blackburn, N., Finite Groups III (Springer-Verlag, New York, 1982).
[10]Kleidman, P.B. and Liebeck, M.W., The subgroup structure of the finite classical groups, London Math. Soc. Lecture Notes (Cambridge University Press, New York, Sydney, 1990).
[11]Li, C.H., ‘On isomorphisms of connected Cayley graphs’, Disc. Math. 78 (1998), 109122.
[12]Li, C.H., ‘Isomorphisms of Cayley digraphs of Abelian groups’, Bull. Austral. Math. Soc. 57 (1998), 181188.
[13]Li, C.H., ‘On isomorphisms of connectd Cayley graphs, II’, J. Combin. Theory (B) (to appear).
[14]Li, C.H., On isomorphisms of finite Cayley graphs – a survey, (UWA Research Report, 1998/8) (University of Western Australia, Australia).
[15]Li, C.H. and Praeger, C.E., ‘The finite simple groups with at most two fusion classes of every order’, Comm. Algebra 24 (1996), 36813704.
[16]Li, C.H., Praeger, C.E. and Xu, M.Y., ‘Isomorphisms of finite Cayley digraphs of bounded valency’, J. Combin. Theory (B) (to appear).
[17]Lorimer, P., ‘Vertex-transitive graphs: symmetric graphs of prime valency’, J. Graph Theory 8(1984), 5568.
[18]Pàlfy, P.P., ‘Isomorphism problem for relational structures with a cyclic automorphism’, European J. Combin. 8 (1987), 3543.
[19]Praeger, C.E., ‘Finite transitive permutation groups and finite vertex-transitive graphs’, in Graph symmetry: algebraic methods and applications 497, NATO Adv. Sc. Inst. Ser.C (Kluwer Acad., Dordrecht, 1997), pp. 277318.
[20]Sabidussi, G.O., ‘Vertex-transitive graphs’, Monatsh. Math. 68 (1964), 426438.
[21]Suzuki, M., Group theory I (Spring-Verlag, Berlin, Heidelberg, New York, 1982).
[22]Xu, M.Y., ‘Half-transitive graphs of prime-cube order’, J. Algebric Combin. 1 (1992), 275282.
[23]Xu, M.Y. and Meng, J.X., ‘Weakly 3-DCI Abelian groups’, Australas. J. Combin. 13 (1996), 4960.
[24]Zhang, J.P., ‘On finite groups all of whose elements of the same order are conjugate in their automorphism groups’, J. Algebra 153 (1992), 2236.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

On isomorphisms of connected Cayley graphs, III

  • Cai Heng Li (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed