Maximal subgroups of symmetric groups
type: Item
instance of: scholarly article
title: Maximal subgroups of symmetric groups
zbMATH Open document ID: 0866.20003
author: Martin W. Liebeck
author: Aner Shalev
published in: Journal of Combinatorial Theory. Series A
publication date: 1997-07-13T00:00:00Z
review text: The purpose of this paper is to give estimates on the number of conjugacy classes of maximal subgroups of the finite symmetric groups \(S_n\), on \(n\) letters in terms of \(n\). It is shown that this number is of the form \((\frac12+o(1))n\). The main work has to be done in establishing that \(S_n\) has at most \(n^{6/11+o(1)}\) conjugacy classes of primitive maximal subgroups. Of course, the O'Nan-Scott Theorem and the classification of finite simple groups are used. In the course of the proof, it is shown that any finite almost simple group has at most \(n^{17/11+o(1)}\) maximal subgroups of index \(n\). For most types of simple groups a slightly stronger estimate has been proved by \textit{A. Mann} and the second author [Isr. J. Math. 96, pt. B, 449-468 (1996)]; the remaining groups of Lie type \(F_4\), \(E^\varepsilon_6\), \(E_7\) and \(E_8\) are dealt with in the paper.
Mathematics Subject Classification ID: 20B35
Mathematics Subject Classification ID: 20E28
Mathematics Subject Classification ID: 20B15
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 11N45
Mathematics Subject Classification ID: 20D06
Mathematics Subject Classification ID: 20D25
zbMATH DE Number: 934986
zbMATH Keywords: number of conjugacy classes of maximal subgroups
zbMATH Keywords: finite symmetric groups
zbMATH Keywords: primitive maximal subgroups
zbMATH Keywords: O'Nan-Scott theorem
zbMATH Keywords: classification of finite simple groups
zbMATH Keywords: finite almost simple groups
zbMATH Keywords: groups of Lie type
MaRDI profile type: Publication
OpenAlex ID: W2057847604
DOI: 10.1006/JCTA.1996.0082
description: scientific article; zbMATH DE number 934986