Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-17T10:16:41.515Z Has data issue: false hasContentIssue false
  • English
  • Français

A 2-ARC TRANSITIVE PENTAVALENT CAYLEY GRAPH OF $\text{A}_{39}$

Part of: Permutation groups Representation theory of groups Graph theory

Published online by Cambridge University Press:  11 January 2016

BO LING  and
BEN GONG LOU
BO LING
Affiliation:
School of Mathematics and Statistics, Yunnan University, Kunmin 650031, PR China email [email protected]
BEN GONG LOU*
Affiliation:
School of Mathematics and Statistics, Yunnan University, Kunmin 650031, PR China email [email protected]
*
✉[email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Zhou and Feng [‘On symmetric graphs of valency five’, Discrete Math. 310 (2010), 1725–1732] proved that all connected pentavalent 1-transitive Cayley graphs of finite nonabelian simple groups are normal. We construct an example of a nonnormal 2-arc transitive pentavalent symmetric Cayley graph on the alternating group $\text{A}_{39}$. Furthermore, we show that the full automorphism group of this graph is isomorphic to the alternating group $\text{A}_{40}$.

Keywords

normal Cayley graphsymmetric graphautomorphism groupfinite simple group

MSC classification

Primary: 20B25: Finite automorphism groups of algebraic, geometric, or combinatorial structures
Secondary: 05C25: Graphs and abstract algebra (groups, rings, fields, etc.) 20D06: Simple groups
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 93 , Issue 3 , June 2016 , pp. 441 - 446
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

References

Bosma, W., Cannon, C. and Playoust, C., ‘The MAGMA algebra system I: the user language’, J. Symbolic Comput. 24 (1997), 235–265.CrossRefGoogle Scholar
Fang, X. G., Li, C. H. and Xu, M. Y., ‘On edge-transitive Cayley graphs of valency four’, European J. Combin. 25 (2004), 1107–1116.Google Scholar
Giudici, M., Li, C. H. and Praeger, C. E., ‘Analysing finite locally s-arc transitive graphs’, Trans. Amer. Math. Soc. 356 (2003), 291–317.Google Scholar
Guo, S. T. and Feng, Y. Q., ‘A note on pentavalent s-transitive graphs’, Discrete Math. 312 (2012), 2214–2216.Google Scholar
Huppert, B., Eudiche Gruppen I (Springer, Berlin, 1967).CrossRefGoogle Scholar
Li, C. H., Isomorphisms of Finite Cayley Graphs, PhD Thesis, The University of Western Australia, 1996.Google Scholar
Li, J. J. and Lu, Z. P., ‘Cubic s-transitive Cayley graphs’, Discrete Math. 309 (2009), 6014–6025.Google Scholar
Praeger, C. E., ‘An O’Nan–Scott theorem for finite quasiprimitive permutation groups and an application to 2-arc-transitive graphs’, J. Lond. Math. Soc. (2) 47 (1992), 227–239.Google Scholar
Robinson, D. J. S., A Course in the Theory of Groups (Springer, New York, 1982).CrossRefGoogle Scholar
Sabidussi, G., ‘On a class of fixed-point-free graphs’, Proc. Amer. Math. Soc. 9 (1958), 800–804.CrossRefGoogle Scholar
Xu, S. J., Fang, X. G., Wang, J. and Xu, M. Y., ‘On cubic s-arc-transitive Cayley graphs of finite simple groups’, European J. Combin. 26 (2005), 133–143.CrossRefGoogle Scholar
Xu, S. J., Fang, X. G., Wang, J. and Xu, M. Y., ‘5-arc transitive cubic Cayley graphs on finite simple groups’, European J. Combin. 28 (2007), 1023–1036.Google Scholar
Zhou, J. X. and Feng, Y. Q., ‘On symmetric graphs of valency five’, Discrete Math. 310 (2010), 1725–1732.Google Scholar