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Maximal subgroups of odd index in finite groups with simple classical socle

Published online by Cambridge University Press:  05 July 2011

N. V. Maslova
Affiliation:
Institute of Mathematics and Mechanics of UB RAS, Russia
C. M. Campbell
Affiliation:
University of St Andrews, Scotland
M. R. Quick
Affiliation:
University of St Andrews, Scotland
E. F. Robertson
Affiliation:
University of St Andrews, Scotland
C. M. Roney-Dougal
Affiliation:
University of St Andrews, Scotland
G. C. Smith
Affiliation:
University of Bath
G. Traustason
Affiliation:
University of Bath
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Summary

Abstract

We discuss the completion of the classification of maximal subgroups of odd index in finite groups with simple classical socle.

Introduction

The subgroup of a finite group G generated by the set of all its minimal non-trivial normal subgroups is called the socle of G and is denoted by Soc(G). A finite group is almost simple if its socle is a nonabelian simple group. It is well known that a finite group G is almost simple if and only if there exists a nonabelian finite simple group L such that L ≃ Inn(L) ⊴ G ≤ Aut(L). In this case Inn(L) = Soc(G). One of the greatest results in the theory of finite permutation groups was obtained by Liebeck and Saxl [7] and independently by Kantor [3]. They gave the classification of finite primitive permutation groups of odd degree. In particular, for each finite group G whose socle is a simple classical group they specified types of subgroups which can be maximal subgroups of odd index in G. However, not every subgroup of these types is a maximal subgroup of odd index in G. Thus, the classification of maximal subgroups of odd index in finite groups with a simple classical socle is not complete. In this paper, we discuss the completion of the classification of maximal subgroups of odd index in finite groups with simple classical socle.

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Publisher: Cambridge University Press
Print publication year: 2011

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References

[1] M., Aschbacher, On the maximal subgroups of the finite classical groups, Invent. Math. 76 (1984), 469–514.Google Scholar
[2] J. H., Conway, R. T., Curtis, S. P., Norton, R. A., Parker, and R. A., Wilson, Atlas of Finite Groups, Clarendon Press, 1985.Google Scholar
[3] W. M., Kantor, Primitive permutation groups of odd degree, and an application to finite projective planes, J. Algebra 106 (1987), 15–45.Google Scholar
[4] P., Kleidman, The subgroup structure of some finite simple groups, Ph.D. Thesis, Cambridge University, 1986.
[5] P., Kleidman, The maximal subgroup structure of the finite 8-dimensional orthogonal groups (q) and of their automorphism groups, J. Algebra 110 (1987), no. 1, 173–242.Google Scholar
[6] P. B., Kleidman and M. W., Liebeck, The subgroup structure of finite classical groups, Cambridge University Press, 1990.Google Scholar
[7] M. W., Liebeck and J., Saxl, The primitive permutation groups of odd degree, J. London Math. Soc. (2) 31 (1985), 250–264.Google Scholar
[8] N. V., Maslova, Classification of maximal subgroups of odd index in finite simple classical groups, Trudy IMM UrO RAN 14 (2008), 4, 100–118 (In Russian). English translation: Proceedings of the Steklov Institute of Mathematics, Suppl. 3 (2009), S164–S183.
[9] N. V., Maslova, Maximal subgroups of odd index in finite groups with simple linear, unitary or symplectic socle, Algebra and Logic, to appear.
[10] J. G., Thompson, Hall subgroups of the symmetric groups, J. Combinatorial Theory 1 (1966), 271–279.Google Scholar

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