Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-26T23:55:21.389Z Has data issue: false hasContentIssue false
  • English
  • Français

Transitive simple subgroups of wreath products in product action

Part of: Permutation groups

Published online by Cambridge University Press:  09 April 2009

Robert W. Baddeley ,
Cheryl E. Praeger  and
Csaba Schneider
Robert W. Baddeley
Affiliation:
32 Arbury Road, Cambridge CB4 2JE, UK e-mail: robert.baddeley@ntlworld.com
Cheryl E. Praeger
Affiliation:
School of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Crawley 6009 WA, Australia e-mail: praeger@maths.uwa.edu.au URL: http://www.maths.uwa.edu.au/~praeger
Csaba Schneider
Affiliation:
Informatics Laboratory, Computer and Automation Research Institute, The Hungarian Academy of Sciences, 1111. Budapest, Lágymányosi u. 11Hungary e-mail: csaba.schneider@sztaki.hu URL: www.sztaki.hu/~schneider
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.

A transitive simple subgroup of a finite symmetric group is very rarely contained in a full wreath product in product action. All such simple permutation groups are determined in this paper. This remarkable conclusion is reached after a definition and detailed examination of ‘Cartesian decompositions’ of the permuted set, relating them to certain ‘Cartesian systems of subgroups’. These concepts, and the bijective connections between them, are explored in greater generality, with specific future applications in mind.

Keywords

permutation groupswreath productsproduct actioninnately transitive groupsplinthCartesian decompositionsCartesian systemsfinite simple groups

MSC classification

Secondary: 20B05: General theory for finite groups 20B15: Primitive groups 20B35: Subgroups of symmetric groups 20B99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 77 , Issue 1 , August 2004 , pp. 55 - 72
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Baddeley, R. W. and Praeger, C. E., ‘On classifying all full factorisations and multiple-factorisations of the finite almost simple groups’, J. Algebra 204 (1998), 129–187.CrossRefGoogle Scholar
[2]Baddeley, R. W., ‘On primitive overgroups of quasiprimitive permutation groups’, J. Algebra 263 (2003), 234–344.CrossRefGoogle Scholar
[3]Baddeley, R. W., Praeger, C. E. and Schneider, C., ‘Innately transitive subgroups of wreath products in product action’, preprint arxiv.org/abs/math.GR/0312352.CrossRefGoogle Scholar
[4]Bamberg, J. and Praeger, C. E., ‘Finite permutation groups with a transitive minimal normal subgroup’, Proc. London Math. Soc., to appear.Google Scholar
[5]Baumeister, Barbara, ‘Factorizations of primitive permutation groups’, J. Algebra 194 (1997), 631–653.CrossRefGoogle Scholar
[6]Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A. and Wilson, R. A., Atlas of finite groups (Oxford University Press, Oxford, 1985).Google Scholar
[7]Dixon, J. D. and Mortimer, B., Permutation groups (Springer, New York, 1996).CrossRefGoogle Scholar
[8]Huppert, B., Endliche Gruppen I (Springer, Berlin, 1967).CrossRefGoogle Scholar
[9]Kleidman, P. B., ‘The maximal subgroups of the finite 8-dimensional orthogonal groups PΩ+8(q) and of their automorphism groups’, J. Algebra 110 (1987), 173–242.CrossRefGoogle Scholar
[10]Kovács, L. G., ‘Wreath decompositions of finite permutation groups’, Bull. Austral. Math. Soc. 40 (1989), 255–279.CrossRefGoogle Scholar
[11]Lidl, R. and Niederreiter, H., Finite fields, 2nd edition (Cambridge University Press, Cambridge, 1997).Google Scholar
[12]Liebeck, M. W., Praeger, C. F. and Saxl, J., ‘A classification of the maximal subgroups of the finite alternating and symmetric groups’, J. Algebra 111 (1987), 365–383.CrossRefGoogle Scholar
[13]Praeger, C. E., ‘The inclusion problem for finite primitive permutation groups’, Proc. London Math. Soc. (3) 60 (1990), 68–88.CrossRefGoogle Scholar