On a generalization of the Hughes subgroup
type: Item
instance of: scholarly article
title: On a generalization of the Hughes subgroup
zbMATH Open document ID: 0941.20012
DOI: 10.1007/BF02675951
published in: Siberian Mathematical Journal
publication date: 2000-04-13T00:00:00Z
review text: By definition, \(G\in{\mathcal H}_p(n)\) whenever \(G\) has \(n\) proper subgroups whose set-theoretic union is not \(G\) and all elements outside the union are of order \(p\). The author extends his previous results [Algebra Logika 19, 118-129 (1980; Zbl 0475.20018), Mat. Sb., Nov. Ser. 130(172), No. 1(5), 120-127 (1986; Zbl 0608.20025), and Sib. Mat. Zh. 30, No. 6(178), 208-218 (1989; Zbl 0727.20015)] on groups in \({\mathcal H}_p(1)\) to the case of \({\mathcal H}_p(n)\), \(n\leq p\). Typical results are as follows: (1) Suppose that \(G\) is a soluble \(p\)-group of derived length \(s\) in \({\mathcal H}_p(n)\) with \(n\leq p\). Then \(G\) has a normal subgroup of index dividing \(p^n\) which is nilpotent of class at most \((p^s-1)/(p-1)\). (2) Suppose that \(G\) is a soluble \(p\)-group of derived length \(s\) of exponent \(p^m\) in \({\mathcal H}_p(n)\) with \(n\leq p\). Then \(G\) is nilpotent of class bounded in terms of \(p\), \(s\), and \(m\).
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20D25
zbMATH DE Number: 1431689
zbMATH Keywords: finite groups
zbMATH Keywords: Hughes subgroups
zbMATH Keywords: nilpotent groups
zbMATH Keywords: derived lengths
zbMATH Keywords: normal subgroups
zbMATH Keywords: nilpotency classes
author: Evgenii I. Khukhro
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1431689