Gradewise properties of subgroups of finite groups.
type: Item
instance of: scholarly article
title: Gradewise properties of subgroups of finite groups.
zbMATH Open document ID: 1337.20010
DOI: 10.1134/S0037446615030027
published in: Siberian Mathematical Journal
publication date: 2015-09-28T00:00:00Z
review text: Let \(G\) be a group, \(A\) a subgroup of \(G\) and let \(\theta\) be a group theoretic property. The subgroup \(A\) enjoys the gradewise property \(\theta\) in \(G\) whenever \(G\) has a normal series \(1=G_0<G_1<\cdots<G_t=G\) such that \((A\cap G_i)G_{i-1}/G_{i-1}\) enjoys property \(\theta\) in \(G_i/G_{i-1}\) for each \(i=1,2,\ldots,t\). In this paper the authors obtain some new characterizations of finite supersolvable and solvable groups. A typical result, among others, is Theorem A: A finite group \(G\) is solvable if and only if some maximal subgroup \(M\) of \(G\) is gradewise nilpotent and the Sylow \(2\)-subgroup of \(M\) is gradewise abelian in \(G\).
reviewed by: Enrico Jabara
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D30
zbMATH DE Number: 6485693
zbMATH Keywords: finite groups
zbMATH Keywords: subgroup functors
zbMATH Keywords: gradewise properties
zbMATH Keywords: finite solvable groups
zbMATH Keywords: finite supersolvable groups
zbMATH Keywords: normal series
author: Wenbin Guo
author: A. N. Skiba
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 6485693