Sylow numbers and nilpotent Hall subgroups.
type: Item
instance of: scholarly article
title: Sylow numbers and nilpotent Hall subgroups.
zbMATH Open document ID: 1285.20019
author: Alexander Moretó
published in: Journal of Algebra
publication date: 2013-11-01T00:00:00Z
full work available at URL: S0021869313000409
reviewed by: Antonio Beltrán
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 6221923
zbMATH Keywords: finite groups
zbMATH Keywords: nilpotent Hall subgroups
zbMATH Keywords: nilpotent subgroups
zbMATH Keywords: Sylow subgroups
zbMATH Keywords: Sylow numbers
MaRDI profile type: Publication
OpenAlex ID: W2013524041
DOI: 10.1016/J.JALGEBRA.2012.12.030
review text: Let \(\pi\) be a set of primes of a finite group \(G\). By the renowned Wielandt's theorem, if \(G\) has a nilpotent Hall \(\pi\)-subgroup then all Hall \(\pi\)-subgroups of \(G\) are conjugate and every \(\pi\)-subgroup is contained in some Hall \(\pi\)-subgroup of \(G\). In particular, all \(\pi\)-subgroups of \(G\) are nilpotent. The author presents a very interesting criterion for a finite group to possess nilpotent Hall \(\pi\)-subgroups. This is done by means of the Sylow numbers.NEWLINENEWLINE The main result claims that \(G\) has nilpotent Hall \(\pi\)-subgroups if and only if the three following conditions hold: (i) For any two different primes \(p,q\in\pi\), the prime \(p\) does not divide the number of Sylow \(q\)-subgroups of \(G\). (ii) If \(\{2,3\}\subseteq\pi\), then \(G\) does not have any composition factor isomorphic to \(\text{PSL}(2,q)\) with \((q^2-1)_{\{2,3\}}=24\). (iii) If \(\{2,7\}\subseteq\pi\), then \(G\) does not have any composition factor isomorphic to \(^2G_2(3^{2n+1})\) with \(n\not\equiv 3\pmod 7\). -- The proof is made first for the case \(|\pi|=2\) and then for the general case, and it depends on the recently completed classification of Hall subgroups of simple groups.
description: scientific article; zbMATH DE number 6221923