Finite groups with some subgroups of Sylow subgroups \(c\)-supplemented
type: Item
instance of: scholarly article
title: Finite groups with some subgroups of Sylow subgroups \(c\)-supplemented
zbMATH Open document ID: 0953.20010
published in: Journal of Algebra
publication date: 2000-11-05T00:00:00Z
review text: The author introduces a generalization of both ``being complemented'' and his concept of \(c\)-normality [J. Algebra 180, No. 3, 954-965 (1996; Zbl 0847.20010)] as follows: a subgroup \(H\) of a group \(G\) is said to be \(c\)-supplemented (in \(G\)) if there exists a subgroup \(K\) of \(G\) such that \(G=HK\) and \(H\cap K\leq\text{core}_G(H)\), the largest normal subgroup of \(G\) contained in \(H\). Theorem 3.1: Let \(G\) be a finite group and let \(P\) be a Sylow \(p\)-subgroup of \(G\) where \(p\) is a prime divisor of \(|G|\) with \((|G|,p-1)=1\). Suppose that every maximal subgroup of \(P\) is \(c\)-supplemented in \(G\) and any two complements of \(P\) in \(G\) are conjugate in \(G\). Then \(G/O_p(G)\) is \(p\)-nilpotent and every \(p'\)-subgroup of \(G\) is contained in some Hall \(p'\)-subgroup of \(G\). Theorem 3.3: Let \(G\) be a finite group and let \(N\) be a normal subgroup of \(G\) such that \(G/N\) is supersoluble. If every maximal subgroup of every Sylow subgroup of \(N\) is \(c\)-supplemented in \(G\), then \(G\) is supersoluble. Theorem 4.2: Let \(G\) be a finite group and let \(p\) be the smallest prime divisor of \(|G|\). If \(G\) is \(A_4\)-free and every second-maximal subgroup of a Sylow \(p\)-subgroup of \(G\) is \(c\)-normal in \(G\), then \(G/O_p(G)\) is \(p\)-nilpotent. The last two theorems generalize results by \textit{A. Ballester-Bolinches} and \textit{X. Guo} [Arch. Math. 72, No. 3, 161-166 (1999; Zbl 0929.20015)].
reviewed by: Hans Lausch
Mathematics Subject Classification ID: 20D40
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D35
zbMATH DE Number: 1428251
zbMATH Keywords: \(c\)-normality
zbMATH Keywords: finite groups
zbMATH Keywords: Sylow subgroups
zbMATH Keywords: maximal subgroups
zbMATH Keywords: complements
zbMATH Keywords: Hall subgroups
zbMATH Keywords: \(c\)-supplemented subgroups
author: Yanming Wang
MaRDI profile type: Publication
full work available at URL: jabr.1999.8079
OpenAlex ID: W2032576239
DOI: 10.1006/JABR.1999.8079
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description: scientific article; zbMATH DE number 1428251