Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The finite vertex-primitive and vertex-biprimitive $s$-transitive graphs for $s\ge4$


Author: Cai Heng Li
Journal: Trans. Amer. Math. Soc. 353 (2001), 3511-3529
MSC (2000): Primary 05C25, 20B05
DOI: https://doi.org/10.1090/S0002-9947-01-02768-4
Published electronically: April 24, 2001
MathSciNet review: 1837245
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A complete classification is given for finite vertex-primitive and vertex-biprimitive $s$-transitive graphs for $s\ge4$. The classification involves the construction of new 4-transitive graphs, namely a graph of valency 14 admitting the Monster simple group $\text{M}$, and an infinite family of graphs of valency 5 admitting projective symplectic groups $\text{PSp}(4,p)$ with $p$ prime and $p\equiv\pm1$ (mod 8). As a corollary of this classification, a conjecture of Biggs and Hoare (1983) is proved.


References [Enhancements On Off] (What's this?)

  • 1. M. Aschbacher, Overgroups of Sylow Subgroups in Sporadic Groups, Memoirs Amer. Math. Soc., 343 (1986). MR 87e:20037
  • 2. M. Aschbacher and G. M. Seitz, Involutions in Chevalley groups over fields of even order, Nagoya Math. J. 63 (1976), 1-91; Correction, Nagoya Math. J. 72 (1978), 135-136. MR 54:10391; MR 80b:20058
  • 3. N. Biggs, Algebraic Graph Theory, Cambridge Univ. Press, New York (2th Ed. 1993). MR 95h:05105
  • 4. N. Biggs and M. Hoare, The sextet construction for cubic graphs, Combinatorica 3 (1983), 153-165. MR 85a:05038
  • 5. A. M. Cohen, M. W. Liebeck, J. Saxl and G. M. Seitz, The local maximal subgroups of the exceptional groups of Lie type, Proc. London Math. Soc. (3), 64 (1992), 21-48. MR 92m:20012
  • 6. M. Conder, An infinite family of 5-arc-transitive cubic graphs, Ars Combin. 25(A) (1988), 95-108. MR 89f:05092
  • 7. M. Conder and C. Walker, The infinitude of 7-arc transitive graphs, J. Algebra 208 (1998), 619-629. MR 99j:05089
  • 8. J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, Atlas of Finite Groups, Clarendon Press, Oxford, 1985. MR 88g:20025
  • 9. A. Gardiner, Arc transitivity in graphs, Quart. J. Math. Oxford (2) 24 (1973), 399-407. MR 48:1973
  • 10. A. Gardiner, Doubly primitive vertex stabilizers in graphs, Math. Z. 135 (1974), 257-266. MR 54:143
  • 11. D. Goldschmidt, Automorphisms of trivalent graphs, Ann. Math. 111 (1980), 377-406. MR 82a:05052
  • 12. D. Gorenstein and R. Lyons, The local structure of finite groups of characteristic 2 type, Memoirs of Amer. Math. Soc., 42 (1983), No. 276. MR 84g:20025
  • 13. A. A. Ivanov and S. V. Shpectorov, Applications of group amalgams to algebraic graph theory, in Investigations in Algebraic Theory of Combinatorial Objects, Ed. by I. Faradzev, 417-441, (Kluwer Academic Publ. 1994). MR 95m:05122
  • 14. P. Kleidman, The maximal subgroups of the Steinberg triviality groups $^3\operatorname{D}_4(q)$ and their automorphism groups, J. Algebra 115 (1988), 182-199. MR 89f:20024
  • 15. P. Kleidman, The maximal subgroups of the Chevalley groups $\operatorname{G} _2(q)$with $q$ odd, the Ree groups $^2\operatorname{G} _2(q)$ and their automorphism groups, J. Algebra 117 (1988), 30-71. MR 89j:20055
  • 16. P. Kleidman and M. Liebeck, The Subgroup Structure of the Finite Classical Groups, London Math. Soc. Lecture Notes, Series 129, Cambridge University Press, Cambridge (1990). MR 91g:20001
  • 17. C. H. Li, A family of quasiprimitive 2-arc-transitive graphs which have non-quasiprimitive full automorphism groups, Europ. J. Combin. 19 (1998), 499-502. MR 99e:05066
  • 18. C. H. Li, Finite $s$-arc transitive graphs of prime-power order, Bull. London Math. Soc. (to appear).
  • 19. C. H. Li, On finite $s$-transitive graphs of odd order, J. Combin. Theory Ser. B, 81 (2001), 307-317.
  • 20. C. H. Li, C. E. Praeger, A. Venkatech and S. Zhou, Finite locally quasiprimitive graphs, Discrete Math. (to appear).
  • 21. M. Liebeck, C. E. Praeger and J. Saxl, The Maximal Factorizations of the Finite Simple Groups and Their Automorphism Groups, Memoirs of Amer. Math. Soc., 86 (1990). MR 90k:20048
  • 22. G. Malle, The maximal subgroups of $^2\operatorname{F}_4(q^2)$, J. Algebra 139 (1991), 52-69. MR 92d:20068
  • 23. U. Meierfrankenfeld and S. V. Shpektorov, The maximal 2-local subgroups of the Monster and Baby Monster, in preparation.
  • 24. C. E. Praeger, An O'Nan-Scott theorem for finite quasiprimitive permutation groups and an application to 2-arc transitive graphs, J. London. Math. Soc. 47 (1992), 227-239. MR 94f:05068
  • 25. C. E. Praeger, On a reduction theorem for finite bipartite 2-arc transitive graphs, Australs. J. Combin. 7 (1993), 21-36. MR 93m:05091
  • 26. C. E. Praeger, Finite transitive permutation groups and finite vertex-transitive graphs, Graph Symmetry: Algebraic Methods and Applications, NATO Ser. C, 497 (1997), pp.277-318. MR 98h:05093
  • 27. G. O. Sabidussi, Vertex-transitive graphs, Monatsh Math. 68 (1964), 426-438. MR 31:91
  • 28. G. Stroth and R. Weiss, A new construction of the group $Ru$, Quart. J. Math. Oxford Ser. (2) 41 (1990), 237-243. MR 91m:20025
  • 29. M. Suzuki, On a class of doubly transitive groups, Ann. Math. 75 (1962), 104-145. MR 25:112
  • 30. M. Suzuki, Group Theory I, Springer-Verlag, New York, 1982. MR 82k:20001c
  • 31. J. Tits, Sur la trialité at certains groupes qui s'en déduisent, Publ. Math. IHES 2 (1959), 14-60.
  • 32. W. T. Tutte, A family of cubical graphs, Proc. Cambridge Phil. Soc. 43 (1947), 459-474. MR 9:97g
  • 33. R. Weiss, Groups with a $(B,N)$-pair and locally transitive graphs, Nagoya Math. J. 74 (1979), 1-21. MR 81j:05067
  • 34. R. Weiss, $s$-transitive graphs, In Algebraic Methods in Graph Theory, Vol. 2 (1981), 827-847. MR 83b:05071
  • 35. R. Weiss, The nonexistence of 8-transitive graphs, Combinatorica 1 (1981), 309-311. MR 84f:05050
  • 36. R. Weiss, A characterization of the group $\hat M_{12}$, Algebras Groups Geom. 2 (1985), 555-563. MR 87k:20010
  • 37. R. Weiss, A characterization and another construction of Janko's group $J_3$, Trans. Amer. Math. Soc. 298 (1986), 621-633. MR 88g:20028
  • 38. R. A. Wilson, The maximal subgroups of the Baby Monster, I, J. Algebra 211 (1999), 1-14. MR 2000b:20016
  • 39. W. J. Wong, Determination of a class of primitive permutation groups, Math. Z. 99 (1967), 235-246. MR 35:5502

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 05C25, 20B05

Retrieve articles in all journals with MSC (2000): 05C25, 20B05


Additional Information

Cai Heng Li
Affiliation: Department of Mathematics and Statistics, The University of Western Australia, Nedlands, WA 6907, Australia
Email: li@maths.uwa.edu.au

DOI: https://doi.org/10.1090/S0002-9947-01-02768-4
Received by editor(s): November 12, 1999
Received by editor(s) in revised form: July 11, 2000
Published electronically: April 24, 2001
Additional Notes: This work forms a part of an ARC project and is supported by an ARC Fellowship
The author is grateful to C.E. Praeger, A.A. Ivanov and R. Weiss for their helpful comments on the work, and to the referee for constructive suggestions
Article copyright: © Copyright 2001 American Mathematical Society

American Mathematical Society