Finite arc transitive Cayley graphs and flagtransitive projective planes
Author:
Cai Heng Li
Journal:
Proc. Amer. Math. Soc. 133 (2005), 3141
MSC (2000):
Primary 20B15, 20B30, 05C25
Published electronically:
July 26, 2004
MathSciNet review:
2085150
Fulltext PDF Free Access
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References 
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Abstract: In this paper, a characterisation is given of finite arc transitive Cayley graphs with . In particular, it is shown that, for any given integer with and , there exists a finite set (maybe empty) of transitive Cayley graphs with such that all transitive Cayley graphs of valency are their normal covers. This indicates that arc transitive Cayley graphs with are very rare. However, it is proved that there exist 4arc transitive Cayley graphs for each admissible valency (a prime power plus one). It is then shown that the existence of a flagtransitive nonDesarguesian projective plane is equivalent to the existence of a very special arc transitive normal Cayley graph of a dihedral group.
 1.
B. Alspach, M. Conder, D. Marusic and M. Y. Xu, A classification of arctransitive circulants, J. Algebraic Combin. 5 (1996), no. 2, 8386. MR 97a:05099
 2.
R. Baddeley, Twoarc transitive graphs and twisted wreath products, J. Algebraic Combin. 2 (1993), 215237. MR 94h:05037
 3.
N. Biggs, Algebraic Graph Theory, Cambridge University Press, 2nd edition, New York, 1992. MR 95h:05105
 4.
P. J. Cameron, Permutation groups, London Mathematical Society Student Texts, 45. Cambridge University Press, Cambridge, 1999. x+220 pp. MR 2001c:20008
 5.
P. J. Cameron, C. E. Praeger, J. Saxl and G. M. Seitz, On the Sims conjecture and distance transitive graphs, Bull. London Math. Soc. 15 (1983), 499506. MR 85g:20006
 6.
P. Dembowski, Finite Geometries, Springer, New York, 1968. MR 38:1597
 7.
W. Feit, Finite projective planes and a question about primes, Proc. Amer. Math. Soc. 108 (1990), 561564. MR 90e:51016
 8.
C. D. Godsil, On the full automorphism group of a graph, Combinatorica 1 (1981), 243256. MR 83a:05066
 9.
R. Guralnick, Subgroups of prime power index in a simple group, J. Algebra 81 (1983), 304311. MR 84m:20007
 10.
A. A. Ivanov and M. E. Iofinova, Biprimitive cubic graphs, in Investigations in the Algebraic Theory of Combinatorial Objects, Math. and its Applications (Soviet Series), Vol. 84 (1993), 459472, Kluwer, Dordrecht, Boston, London. MR 88m:05048
 11.
A. A. Ivanov and C. E. Praeger, On finite affine 2arc transitive graphs, Europ. J. Combin. 14 (1993), 421444. MR 94k:05089
 12.
W. M. Kantor, Primitive groups of odd degree and an application to finite projective planes, J. Algebra, 106 (1987), 1545. MR 88b:20007
 13.
C. H. Li, Finite arc transitive graphs of primepower order, Bull. London Math. Soc. 33 (2001), 129137. MR 2002d:05064
 14.
C. H. Li, The finite vertexprimitive and vertexbiprimitive transitive graphs with , Trans. Amer. Math. Soc. 353 (2001), 35113529. MR 2002c:05084
 15.
C. H. Li, On finite 2arctransitive Cayley graphs, submitted.
 16.
C. H. Li and A. Seress, Finite quasiprimitive arc transitive graphs of product action type, in preparation.
 17.
H. van Maldeghem, Generalized Polygons, Birkhäuser Verlag, Boston, Berlin, 1998. MR 2000k:51004
 18.
D. Marusic, On 2arctransitivity of Cayley graphs, J. Combin. Theory Ser. B 87 (2003), no. 1, 162196. MR 2004a:05064
 19.
C. E. Praeger, An O'NanScott theorem for finite quasiprimitive permutation groups and an application to 2arc transitive graphs, J. London. Math. Soc. 47 (1992), 227239. MR 94f:05068
 20.
C. E. Praeger, On a reduction theorem for finite bipartite 2arc transitive graphs, Australas. J. Combin. 7 (1993), 2136. MR 93m:05091
 21.
C. E. Praeger, Finite normal edgetransitive Cayley graphs, Bull. Austral. Math. Soc. 60 (1999), 207220. MR 2000j:05057
 22.
Koen Thas, Finite flagtransitive projective planes: a survey and some remarks, Discrete Math. 266 (2003), 417429.
 23.
V. I. Trofimov and R. M. Weiss, Graphs with a locally linear group of automorphisms, Math. Proc. Cambridge Philos. Soc. 118 (1995), 191206. MR 97d:05143
 24.
R. M. Weiss, The nonexistence of 8transitive graphs, Combinatorica 1 (1981), 309311. MR 84f:05050
 25.
M. Y. Xu, Automorphism groups and isomorphisms of Cayley digraphs, Discrete Math. 182 (1998), 309320. MR 98i:05096
 1.
 B. Alspach, M. Conder, D. Marusic and M. Y. Xu, A classification of arctransitive circulants, J. Algebraic Combin. 5 (1996), no. 2, 8386. MR 97a:05099
 2.
 R. Baddeley, Twoarc transitive graphs and twisted wreath products, J. Algebraic Combin. 2 (1993), 215237. MR 94h:05037
 3.
 N. Biggs, Algebraic Graph Theory, Cambridge University Press, 2nd edition, New York, 1992. MR 95h:05105
 4.
 P. J. Cameron, Permutation groups, London Mathematical Society Student Texts, 45. Cambridge University Press, Cambridge, 1999. x+220 pp. MR 2001c:20008
 5.
 P. J. Cameron, C. E. Praeger, J. Saxl and G. M. Seitz, On the Sims conjecture and distance transitive graphs, Bull. London Math. Soc. 15 (1983), 499506. MR 85g:20006
 6.
 P. Dembowski, Finite Geometries, Springer, New York, 1968. MR 38:1597
 7.
 W. Feit, Finite projective planes and a question about primes, Proc. Amer. Math. Soc. 108 (1990), 561564. MR 90e:51016
 8.
 C. D. Godsil, On the full automorphism group of a graph, Combinatorica 1 (1981), 243256. MR 83a:05066
 9.
 R. Guralnick, Subgroups of prime power index in a simple group, J. Algebra 81 (1983), 304311. MR 84m:20007
 10.
 A. A. Ivanov and M. E. Iofinova, Biprimitive cubic graphs, in Investigations in the Algebraic Theory of Combinatorial Objects, Math. and its Applications (Soviet Series), Vol. 84 (1993), 459472, Kluwer, Dordrecht, Boston, London. MR 88m:05048
 11.
 A. A. Ivanov and C. E. Praeger, On finite affine 2arc transitive graphs, Europ. J. Combin. 14 (1993), 421444. MR 94k:05089
 12.
 W. M. Kantor, Primitive groups of odd degree and an application to finite projective planes, J. Algebra, 106 (1987), 1545. MR 88b:20007
 13.
 C. H. Li, Finite arc transitive graphs of primepower order, Bull. London Math. Soc. 33 (2001), 129137. MR 2002d:05064
 14.
 C. H. Li, The finite vertexprimitive and vertexbiprimitive transitive graphs with , Trans. Amer. Math. Soc. 353 (2001), 35113529. MR 2002c:05084
 15.
 C. H. Li, On finite 2arctransitive Cayley graphs, submitted.
 16.
 C. H. Li and A. Seress, Finite quasiprimitive arc transitive graphs of product action type, in preparation.
 17.
 H. van Maldeghem, Generalized Polygons, Birkhäuser Verlag, Boston, Berlin, 1998. MR 2000k:51004
 18.
 D. Marusic, On 2arctransitivity of Cayley graphs, J. Combin. Theory Ser. B 87 (2003), no. 1, 162196. MR 2004a:05064
 19.
 C. E. Praeger, An O'NanScott theorem for finite quasiprimitive permutation groups and an application to 2arc transitive graphs, J. London. Math. Soc. 47 (1992), 227239. MR 94f:05068
 20.
 C. E. Praeger, On a reduction theorem for finite bipartite 2arc transitive graphs, Australas. J. Combin. 7 (1993), 2136. MR 93m:05091
 21.
 C. E. Praeger, Finite normal edgetransitive Cayley graphs, Bull. Austral. Math. Soc. 60 (1999), 207220. MR 2000j:05057
 22.
 Koen Thas, Finite flagtransitive projective planes: a survey and some remarks, Discrete Math. 266 (2003), 417429.
 23.
 V. I. Trofimov and R. M. Weiss, Graphs with a locally linear group of automorphisms, Math. Proc. Cambridge Philos. Soc. 118 (1995), 191206. MR 97d:05143
 24.
 R. M. Weiss, The nonexistence of 8transitive graphs, Combinatorica 1 (1981), 309311. MR 84f:05050
 25.
 M. Y. Xu, Automorphism groups and isomorphisms of Cayley digraphs, Discrete Math. 182 (1998), 309320. MR 98i:05096
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Additional Information
Cai Heng Li
Affiliation:
School of Mathematics and Statistics, The University of Western Australia, Crawley, 6009 Western Australia, Australia
Email:
li@maths.uwa.edu.au
DOI:
http://dx.doi.org/10.1090/S0002993904075495
PII:
S 00029939(04)075495
Received by editor(s):
August 27, 2003
Received by editor(s) in revised form:
September 11, 2003, and September 24, 2003
Published electronically:
July 26, 2004
Additional Notes:
This work was supported by an Australian Research Council Discovery Grant, and a QEII Fellowship. The author is grateful to the referee for his constructive comments.
Communicated by:
John R. Stembridge
Article copyright:
© Copyright 2004 American Mathematical Society
