Number of Sylow subgroups and \(p\)-nilpotence of finite groups
type: Item
instance of: scholarly article
title: Number of Sylow subgroups and \(p\)-nilpotence of finite groups
zbMATH Open document ID: 0932.20016
author: Naoki Chigira
published in: Journal of Algebra
publication date: 2000-03-09T00:00:00Z
review text: Let \(G\) be a finite group and let \(\pi(G)\) be the set of primes dividing the order of \(G\). The classical theorems of Sylow assert that the number \(n_p(G)\) of Sylow \(p\)-subgroups is congruent to 1 modulo \(p\), which is equal to the index \(|G:N_G(P)|\) for some Sylow \(p\)-subgroup \(P\). A finite group \(G\) is said to be \(p\)-nilpotent if \(G\) has a normal \(p\)-complement for some \(P\in\text{Syl}_p(G)\), the set of Sylow \(p\)-subgroups of \(G\). It is proved that if \(p\neq 3\) is a prime then \(G\) is \(p\)-nilpotent if and only if \((p,n_r(G))=1\) for every \(r\in\pi(G)\) and if \(G\) does not have a composition factor isomorphic with \(U_3(2^q)\), \(q\) even not divisible by 3, then \(G\) is \(p\)-nilpotent if and only if \((p,n_r(G))=1\) for every \(r\in\pi(G)\).
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 1211655
zbMATH Keywords: \(p\)-nilpotence
zbMATH Keywords: numbers of Sylow subgroups
zbMATH Keywords: finite groups
zbMATH Keywords: Sylow theorems
zbMATH Keywords: normal \(p\)-complements
reviewed by: N. M. Kurnosenko
MaRDI profile type: Publication
full work available at URL: jabr.1997.7268
OpenAlex ID: W1965602179
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DOI: 10.1006/JABR.1997.7268
description: scientific article; zbMATH DE number 1211655