Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-10T13:45:58.124Z Has data issue: false hasContentIssue false
  • English
  • Français

OVERGROUPS OF PRIMITIVE GROUPS

Part of: Representation theory of groups Permutation groups

Published online by Cambridge University Press:  01 August 2009

MICHAEL ASCHBACHER
MICHAEL ASCHBACHER*
Affiliation:
California Institute of Technology, Pasadena, California 91125, USA (email: asch@its.caltech.edu)
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.

We give a qualitative description of the set 𝒪G(H) of overgroups in G of primitive subgroups H of finite alternating and symmetric groups G, and particularly of the maximal overgroups. We then show that certain weak restrictions on the lattice 𝒪G(H) impose strong restrictions on H and its overgroup lattice.

Keywords

permutation grouplattice

MSC classification

Secondary: 20B15: Primitive groups 20D30: Series and lattices of subgroups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 87 , Issue 1 , August 2009 , pp. 37 - 82
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2009

References

[1]Aschbacher, M., ‘Overgroups of Sylow subgroups in sporadic groups’, Mem. Amer. Math. Soc. 60(343) (1985), 1–235.Google Scholar
[2]Aschbacher, M., Finite Group Theory (Cambridge University Press, Cambridge, 1986).Google Scholar
[3]Aschbacher, M., ‘On intervals in subgroup lattices of finite groups’, J. Amer. Math. Soc. 21 (2008), 809–830.CrossRefGoogle Scholar
[4]Aschbacher, M., ‘Signalizer lattices in finite groups’, Michigan Math. J. 58(1) (2009), 79–103.CrossRefGoogle Scholar
[5]Aschbacher, M. and Scott, L., ‘Maximal subgroups of finite groups’, J. Algebra 92 (1985), 44–80.CrossRefGoogle Scholar
[6]Baddeley, R. and Lucchini, A., ‘On representing finite lattices as intervals in subgroup lattices of finite groups’, J. Algebra 196 (1997), 1–100.CrossRefGoogle Scholar
[7]Baddeley, R. and Praeger, C., ‘On primitive overgroups of quasiprimitive permutation groups’, J. Algebra 263 (2003), 294–344.CrossRefGoogle Scholar
[8]Baddeley, R., Praeger, C. and Schneider, C., ‘Transitive simple subgroups of wreath products in product action’, J. Aust. Math. Soc. 77 (2004), 55–72.CrossRefGoogle Scholar
[9]Baddeley, R., Praeger, C. and Schneider, C., ‘Innately transitive subgroups of wreath products in product action’, Trans. Amer. Math. Soc. 358 (2006), 1619–1641.CrossRefGoogle Scholar
[10]Baddeley, R., Praeger, C. and Schneider, C., ‘Quasiprimitive groups and blow-up decompositions’, J. Algebra 311 (2007), 337–351.CrossRefGoogle Scholar
[11]Gorenstein, D., Lyons, R. and Solomon, R., The Classification of the Finite Simple Groups, Number 3, Mathematical Surveys and Mongraphs, 40 (American Mathematical Society, Providence, RI, 1999).CrossRefGoogle Scholar
[12]Guralnick, R., ‘Subgroups of prime power index in a simple group’, J. Algebra 81 (1983), 304–311.CrossRefGoogle Scholar
[13]Kleidman, P., ‘The maximal subgroups of the 8-dimensional orthogonal groups PΩ 8+(q) and of their automorphism groups’, J. Algebra 110 (1987), 173–242.CrossRefGoogle Scholar
[14]Liebeck, M., Praeger, C. and Saxl, J., ‘A classification of the maximal subgroups of the finite alternating and symmetric groups’, J. Algebra 111 (1987), 365–383.CrossRefGoogle Scholar
[15]Liebeck, M., Prager, C. and Saxl, J., ‘The maximal factorizations of the finite simple groups and their automorphism groups’, Mem. Amer. Math. Soc. 86(432) (1990), 1–151.Google Scholar
[16]Niven, I. and Zuckerman, H., An Introduction to the Theory of Numbers (Wiley, New York, 1980).Google Scholar
[17]Palfy, P. and Pudlak, P., ‘Congruence lattices of finite algebras and intervals in subgroup lattices of finite groups’, Algebra Universalis 61 (1980), 22–27.CrossRefGoogle Scholar
[18]Praeger, C., ‘The inclusion problem for finite primitive permutation groups’, Proc. London Math. Soc. 60 (1990), 68–88.CrossRefGoogle Scholar
[19]Tits, J., ‘A local approach to buildings’, in: The Geometric Vein. The Coxeter Festschrift (Springer, New York, 1982), pp. 519–547.Google Scholar