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Bounds and quotient actions of innately transitive groups

Part of: Permutation groups

Published online by Cambridge University Press:  09 April 2009

John Bamberg
John Bamberg
Affiliation:
School of Mathematics and StatisticsUniversity of Western Australia35 Stirling HighwayCrawley WA 6009Australia e-mail: john.bam@maths.uwa.edu.au
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Abstract

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Finite innately transitive permutation groups include all finite quasiprimitive and primitive permutation groups. In this paper, results in the theory of quasiprimitive and primitive groups are generalised to innately transitive groups, and in particular, we extend results of Praeger and Shalev. Thus we show that innately transitive groups possess parameter bounds which are similar to those for primitive groups. We also classify the innately transitive types of quotient actions of innately transitive groups.

MSC classification

Secondary: 20B05: General theory for finite groups 20B35: Subgroups of symmetric groups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 79 , Issue 1 , August 2005 , pp. 95 - 112
Copyright
Copyright © Australian Mathematical Society 2005

References

[1]Bamberg, J. and Praeger, C. E., ‘Finite permutation groups with a transitive minimal normal subgroup’, Proc. London Math. Soc. (3) 89 (2004), 71–103.CrossRefGoogle Scholar
[2]Bochert, A., ‘Ueber die Zahl der verschiedenen Werthe, die eine Function gegebener Buchstaben durch Vertauschung derselben erlangen kann’, Math. Ann. 65 (1889), 584–590.CrossRefGoogle Scholar
[3]Dixon, J. D. and Mortimer, B., Permutation groups (Springer, New York, 1996).CrossRefGoogle Scholar
[4]Heath-Brown, D. R., Praeger, C. E. and Shalev, A., ‘Permutation groups, simple groups, and sieve methods’, Israel J. Math., to appear.Google Scholar
[5]Liebeck, M. W., Praeger, C. E. and Saxl, J., ‘On the O'Nan-Scott theorem for finite primitive permutation groups’, J. Ausiral. Math. Soc. 44 (1988), 389–396.Google Scholar
[6]Praeger, C. E., ‘An O'Nan-Scott theorem for finite quasiprimitive permutation groups and an application to 2-arc transitive graphs’, J. London Math. Soc. (2) 47 (1993), 227–239.CrossRefGoogle Scholar
[7]Praeger, C. E., ‘Quotients and inclusions of finite quasiprimitive permutation groups’, J. Algebra 269 (2003), 329–346.CrossRefGoogle Scholar
[8]Praeger, C. E., Li, C. H. and Niemeyer, A. C., ‘Finite transitive permutation groups and finite vertextransitive graphs’, in: Graph symmetry (Montreal, PQ, 1996) (Kluwer Acad. Publ., Dordrecht, 1997) pp. 277–318.Google Scholar
[9]Praeger, C. E. and Saxl, J., ‘On the orders of primitive permutation groups’, Bull. London Math. Soc. 12 (1980), 303–307.CrossRefGoogle Scholar
[10]Praeger, C. E. and Shalev, A., ‘Bounds on finite quasiprimitive permutation groups’, J. Austral. Math. Soc. 71 (2001), 243–258.CrossRefGoogle Scholar