Maximal supersoluble subgroups of symmetric groups
type: Item
instance of: scholarly article
title: Maximal supersoluble subgroups of symmetric groups
zbMATH Open document ID: 0793.20003
DOI: 10.1007/BF01766310
author: Mariagrazia Bianchi
author: Anna Gillio Berta Mauri
author: Peter Hauck
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 1992-09-27T00:00:00Z
review text: All maximal supersoluble subgroups of symmetric groups are classified. But, in fact, the main result in this paper is the classification of the maximal supersoluble transitive subgroups of the symmetric group \(S_ n\) on \(n\) letters. The proof of the main result splits into two parts. In Section 4 the authors present a certain general construction of maximal supersoluble transitive subgroups of \(S_ n\) and in Section 5 it is shown that there are no others, that is, any maximal supersoluble transitive subgroup of \(S_ n\) is conjugate to one of the subgroups presented in Section 4. Then, the intransitive case can be dealt with by a straightforward reduction to the transitive case to yield the following theorem (cf. Section 7): Let \(n = n_ 1 + \dots + n_ b\) with \(b > 1\) and let \(H = H_ 1 \times H_ 2 \times \dots \times H_ b\), where \(H_ i\) is a maximal supersoluble transitive subgroup of \(S_{n_ i}\); furthermore, set \(\varphi(x) = \prod^ b_{i = 1} (x - n_ i)\). Then \(H\) is a maximal supersoluble subgroup of \(S_ n\) if and only if \((x- 1)(x-2)\nmid \varphi(x)\) and \((x-2^ f)^ 2 \nmid \varphi(x)\) for all \(f\geq 0\). The authors also obtain several other results which are not mentioned here due to the limited space. Suprunenko proved that every symmetric group \(S_ n\) possesses a unique conjugacy class of maximal nilpotent transitive subgroups. This result is important for the work reviewed.
reviewed by: Yongcai Ren
Mathematics Subject Classification ID: 20B35
Mathematics Subject Classification ID: 20B30
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20E28
zbMATH DE Number: 59782
zbMATH Keywords: symmetric groups
zbMATH Keywords: maximal supersoluble transitive subgroups
zbMATH Keywords: construction
zbMATH Keywords: conjugacy class
zbMATH Keywords: maximal nilpotent transitive subgroups
MaRDI profile type: Publication
cites work: Q5182339
cites work: Endliche Gruppen I
cites work: Q4078234
description: scientific article; zbMATH DE number 59782