Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-25T08:15:55.187Z Has data issue: false hasContentIssue false
  • English
  • Français

On the normal structure of a one-point stabilizer in a doubly transitive permutation group

Part of: Permutation groups

Published online by Cambridge University Press:  09 April 2009

M. D. Atkinson  and
Cheryl E. Praeger
M. D. Atkinson
Affiliation:
Department of Computing Mathematics University College Cardiff, Wales
Cheryl E. Praeger
Affiliation:
Department of Mathematics University of Western AustraliaNedlands, Australia6009
Rights & Permissions [Opens in a new window]

Extract

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.

Let G be a doubly transitive permutation group on a finite set Ω, and let Kα be a normal subgroup of the stabilizer Gα of a point α in Ω. If the action of Gα on the set of orbits of Kα in Ω − {α} is 2-primitive with kernel Kα it is shown that either G is a normal extension of PSL(3, q) or Kα ∩ Gγ is a strongly closed subgroup of Gαγ in Gα, where γ ∈ Ω − {α}. If in addition the action of Gα on the set of orbits of Kα is assumed to be 3-transitive, extra information is obtained using permutation theoretic and centralizer ring methods. In the case where Kα has three orbits in Ω − {α} strong restrictions are obtained on either the structure of G or the degrees of certain irreducible characters of G. Subject classification (Amer. Math. Soc. (MOS) 1970: 20 B 20, 20 B 25.

MSC classification

Secondary: 20B20: Multiply transitive finite groups 20B25: Finite automorphism groups of algebraic, geometric, or combinatorial structures
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 25 , Issue 2 , March 1978 , pp. 145 - 166
Copyright
Copyright © Australian Mathematical Society 1978

References

REFERENCES

Atkinson, M. D. (1975), “Doubly transitive but not doubly primitive permutation groups II”, J. London Math. Soc. (2) 10, 53–60.CrossRefGoogle Scholar
Cameron, P. J. (1971), Structure of suborbits in some primitive permutation groups (Thesis, Oxford University).Google Scholar
Cameron, P. J. (1973), “Extending symmetric designs”, J. Combinatorial Theory A 14, 215–220.CrossRefGoogle Scholar
Dembowski, P. (1968), Finite Geometries (Ergebnisse der Math. 44, Springer-Verlag, Berlin-Heidelberg-New York).CrossRefGoogle Scholar
Frame, J. S. (1941), “The double cosets of a finite group”, Bull. Amer. Math. Soc. 47, 458–467.Google Scholar
Gorenstein, D. (1968), Finite Groups (Harper and Row, New York-Evanston-London).Google Scholar
Hale, M. P. Jr and Shult, E. E. (1974), “Equiangular lines, the graph extension theorem and transfer in triply transitive groups”, Math. Z. 135, 111–123.CrossRefGoogle Scholar
Hering, C., Kantor, W. M. and Seitz, G. M. (1972), “Finite groups having a split BN-pair of rank 1”, J. Algebra 20, 435–475.Google Scholar
Higman, D. G. (1971), “Solvability of a class of rank 3 permutation groups”, Nagoya Math. J. 41, 89–96.CrossRefGoogle Scholar
Higman, D. G. (1972), Combinatorial Considerations about Permutation Groups (Lecture Notes, Mathematical Institute, Oxford).Google Scholar
Kantor, W. M. (1974), “2-Transitive designs”, in Combinatorics, Part 3, Combinatorial Group Theory (Proceedings of the Advanced Study Institute on Combinatorics, Breukelen, The Netherlands; ed. Hall, M. Jr and van Lint, J. H., Math. Centre Tracts 57, Amsterdam), pp. 44–97.Google Scholar
Mordell, L. J. (1969), Diophantine Equations (Academic Press, London-New York).Google Scholar
Norman, C. W. (1968), “A characterization of the Mathieu group M11”, Math. Z. 106, 162–166.CrossRefGoogle Scholar
O'Nan, M. E. (1975), “Normal structure of the one-point stabilizer of a doubly-transitive permutation group I”, Trans. Amer. Math. Soc. 214, 1–42;Google Scholar
Normal structure of the one-point stabilizer of a doubly-transitive permutation group II, Trans. Amer. Math. Soc. 214, 43–74.Google Scholar
Ostrom, I. G. and Wagner, A. (1959), “On projective and affine planes with transitive collineation groups”, Math. Z. 71, 186–199.Google Scholar
Praeger, C. E. (1977), “Doubly transitive permutation groups which are not doubly primitive”, J. Algebra 44, 389–395.Google Scholar
Praeger, C. E. (1978), “Doubly transitive permutation groups in which the one point stabiliser is triply transitive on a set of blocks”, J. Algebra (to appear).CrossRefGoogle Scholar
Seidel, J. J. (1973), “A survey of two-graphs”, Proc. Int. Coll. Theorie Combinatorie Acc. Naz. Lincei (Rome).Google Scholar
Wielandt, H. (1964), Finite Permutation Groups (Academic Press, London-New York).Google Scholar