Hostname: page-component-848d4c4894-sjtt6 Total loading time: 0 Render date: 2024-06-15T17:33:16.106Z Has data issue: false hasContentIssue false
  • English
  • Français

Primitive permutation groups with a doubly transitive subconstituent

Part of: Permutation groups Graph theory

Published online by Cambridge University Press:  09 April 2009

Cheryl E. Praeger
Cheryl E. Praeger
Affiliation:
Department of MathematicsThe University of Western AustraliaNedlands, Western Australia 6009, Australia
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.

Let Gbe a primitive permutation group on a finite set Ω. We investigate the subconstitutents of G, that is the permutation groups induced by a point stabilizer on its orbits in Ω, in the cases where Ghas a diagonal action or a product action on Ω. In particular we show in these cases that no subconstituent is doubly transitive. Thus if G has a doubly transitive subconstituent we show that G has a unique minimal normal subgroup N and either N is a nonabelian simple group or N acts regularly on Ω: we investigate further the case where N is regular on Ω.

MSC classification

Secondary: 20B15: Primitive groups 05C25: Graphs and abstract algebra (groups, rings, fields, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 45 , Issue 1 , August 1988 , pp. 66 - 77
Copyright
Copyright © Australian Mathematical Society 1988

References

[1]Aschbacher, M. and Scott, L. L., ‘Maximal subgroups of finite groups’, J. Algebra 92 (1985), 44–80.Google Scholar
[2]Cameron, P. J., ‘Permutation groups with multiply transitive suborbits I’, Proc. London Math. Soc. (3) 25 (1972), 427–440; II Bull. London Math. Soc. 6 (1974), 1–5.CrossRefGoogle Scholar
[3]Cameron, P. J., ‘Finite permutation groups and finite simple groups’, Bull. London Math. Soc. 13 (1981), 1–22.CrossRefGoogle Scholar
[4]Cameron, P. J. and Praeger, C. E., ‘Graphs and permutation groups with projective subconstitutents’, J. London Math. Soc. (2) 25 (1982), 62–74.Google Scholar
[5]Glauberman, G., ‘Central elements in core-free groups’, J. Algebra 4 (1966), 403–420.Google Scholar
[6]Gorenstein, D., Finite groups (Harper and Row, New York, Evanston and London, 1968).Google Scholar
[7]Kovacs, L. G., ‘Maximal subgroups in composite groups’, J. Algebra, 99 (1986), 114–131.Google Scholar
[8]Praeger, C. E., ‘On primitive permutation groups with a doubly transitive suborbit’, J. London Math. Soc. (2) 17 (1978), 67–73.Google Scholar
[9]Praeger, C. E., ‘Primitive permutation groups and a characterization of the odd graphs’, J. Combin. Theory Ser. B 31 (1981), 117–142.Google Scholar
[10]Praeger, C. E., Saxl, J., and Yokoyama, K., ‘Distance transitive graphs and finite simple groups’, Proc. London Math. Soc. (3) 55 (1987), 1–21.CrossRefGoogle Scholar
[11]Scott, L. L., ‘Representations in characteristic p’, Proc. Sympos. Pure Math. 37 (1980), 319–331.CrossRefGoogle Scholar
[12]Wielandt, H., Finite permutation groups (Academic Press, New York, 1964).Google Scholar