Maximal subgroups of symmetric groups

Martin W. Liebeck; Aner Shalev

doi:10.1006/jcta.1996.0082

Maximal subgroups of symmetric groups

Martin W. Liebeck^*, Aner Shalev

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

22 Scopus citations

Abstract

We show that S_n has at most n^{6/11 + o(1)} conjugacy classes of primitive maximal subgroups. This improves an nclog³n bound given by Babai. We conclude that the number of conjugacy classes of maximal subgroups of S_n is of the form (1/2 + o(1))n. It also follows that, for large n, S_n has less than n! maximal subgroups. This confirms a special case of a conjecture of Wall. Improving a recent result from [MSh], we also show that any finite almost simple group has at most n^17/11+o(1) maximal subgroups of index n.

Original language	English
Pages (from-to)	341-352
Number of pages	12
Journal	Journal of Combinatorial Theory. Series A
Volume	75
Issue number	2
DOIs	https://doi.org/10.1006/jcta.1996.0082
State	Published - Aug 1996

Bibliographical note

Funding Information:
The second author thanks the department of mathematics of the University of Chicago for its support and hospitality while this work was carried out.

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10.1006/jcta.1996.0082

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abstract = "We show that Sn has at most n6/11 + o(1) conjugacy classes of primitive maximal subgroups. This improves an nclog3n bound given by Babai. We conclude that the number of conjugacy classes of maximal subgroups of Sn is of the form (1/2 + o(1))n. It also follows that, for large n, Sn has less than n! maximal subgroups. This confirms a special case of a conjecture of Wall. Improving a recent result from [MSh], we also show that any finite almost simple group has at most n17/11+o(1) maximal subgroups of index n.",

author = "Liebeck, {Martin W.} and Aner Shalev",

note = "Funding Information: The second author thanks the department of mathematics of the University of Chicago for its support and hospitality while this work was carried out.",

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AU - Liebeck, Martin W.

AU - Shalev, Aner

N1 - Funding Information: The second author thanks the department of mathematics of the University of Chicago for its support and hospitality while this work was carried out.

PY - 1996/8

Y1 - 1996/8

N2 - We show that Sn has at most n6/11 + o(1) conjugacy classes of primitive maximal subgroups. This improves an nclog3n bound given by Babai. We conclude that the number of conjugacy classes of maximal subgroups of Sn is of the form (1/2 + o(1))n. It also follows that, for large n, Sn has less than n! maximal subgroups. This confirms a special case of a conjecture of Wall. Improving a recent result from [MSh], we also show that any finite almost simple group has at most n17/11+o(1) maximal subgroups of index n.

AB - We show that Sn has at most n6/11 + o(1) conjugacy classes of primitive maximal subgroups. This improves an nclog3n bound given by Babai. We conclude that the number of conjugacy classes of maximal subgroups of Sn is of the form (1/2 + o(1))n. It also follows that, for large n, Sn has less than n! maximal subgroups. This confirms a special case of a conjecture of Wall. Improving a recent result from [MSh], we also show that any finite almost simple group has at most n17/11+o(1) maximal subgroups of index n.

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Maximal subgroups of symmetric groups

Abstract

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