Sylow numbers of finite groups
type: Item
instance of: scholarly article
title: Sylow numbers of finite groups
zbMATH Open document ID: 0832.20042
published in: Journal of Algebra
publication date: 1996-02-25T00:00:00Z
review text: For a finite group \(G\) and a prime \(p\) dividing the order of \(G\), the Sylow \(p\)-number of \(G\) is \(n_p=|G:N_G(P)|\), where \(P\) is any Sylow \(p\)-subgroup of \(G\). The study of these numbers was initiated by \textit{Ph. Hall} [J. Lond. Math. Soc. 3, 98-105 (1928; JFM 54.0145.01)] and continued by \textit{M. Hall jun.} [J. Algebra 7, 363-371 (1967; Zbl 0178.02102)]. The elegant paper under review is the first post-classification attempt to revisit the Sylow numbers; induction and the classification of the simple groups are used to prove in Section 2 the following beautiful arithmetic criterion for \(p\)-nilpotency: A finite group \(G\) is \(p\)-nilpotent for some prime \(p\) if and only if \((p,n_q)=1\) for all prime divisors \(q\) of the order of \(G\). The finite groups in which the Sylow numbers are square-free and those in which the Sylow numbers are prime powers are classified in Sections 4 and 3, respectively. As a consequence, it follows for example that a solvable group with square-free Sylow numbers must be supersolvable. In the last section, the nilpotent length, the derived length and the \(p\)-length of a finite solvable group are bounded in terms of certain invariants related to the Sylow numbers of the group.
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D05
zbMATH DE Number: 802371
zbMATH Keywords: supersolvable groups
zbMATH Keywords: classification of simple groups
zbMATH Keywords: Sylow subgroups
zbMATH Keywords: Sylow numbers
zbMATH Keywords: \(p\)-nilpotency
zbMATH Keywords: finite groups
zbMATH Keywords: solvable groups
zbMATH Keywords: nilpotent length
zbMATH Keywords: derived length
zbMATH Keywords: \(p\)-length
author: Jiping Zhang
reviewed by: Marian Deaconescu
MaRDI profile type: Publication
full work available at URL: jabr.1995.1235
OpenAlex ID: W2066826922
DOI: 10.1006/JABR.1995.1235
description: scientific article; zbMATH DE number 802371