Direct decompositions of ideals and normal subgroups
type: Item
instance of: scholarly article
title: Direct decompositions of ideals and normal subgroups
zbMATH Open document ID: 0638.20021
DOI: 10.1016/0001-8708(88)90005-9
author: Hartmut Laue
published in: Advances in Mathematics
publication date: 1988-01-01T00:00:00Z
review text: Using some rather unusual techniques (involving a careful analysis of the action of certain sets of operators on subjacent subgroups of skew rings), the author obtains two quite general results (Theorems (2.5) and (3.2)) on direct decompositions of groups. Their statements are too technical to be reproduced here; however, to give an idea of what happens, note that by suitable choices of the set of operators, Maschke's problem and a result of Nakayama [Th. VII 9.3 of ``Finite Groups II'' (1982; Zbl 0477.20001) by \textit{B. Huppert} and \textit{N. Blackburn}] are obtained as particular cases, as well as the classical Krull-Schmidt theorem. Due to the techniques used, these results apply to groups as well as to rings, their particular versions for algebras over fields being of interest from the representation point of view. We state here only two of their corollaries, in their version for groups. Theorem I. Let G be a finite group, H an abelian normal subgroup of G and \(W\trianglelefteq X\leq N_{Aut(G)}(H)\) such that \((| X/W|,| H|)=1\). Then among the maximal W-invariant normal subgroups of G contained in H and having a W-invariant complement in G there is one which is X-invariant and has an X-invariant complement in G. Theorem II. Let G be a finite group and \(W\trianglelefteq X\leq Aut(G)\) such that \((| X/W|,| G/G'|,| Z(G)|)=1\). Then: i) G is the X-direct product of an abelian normal subgroup and a normal subgroup without any W-invariant direct abelian nontrivial factor. ii) If G is X-direct indecomposable, then G is the direct product of isomorphic W-invariant and W-indecomposable subgroups. If, moreover, G is nonabelian, then there exists a direct decomposit\({}_ 0\) which label \(\alpha\). The entire lattice \({\mathcal L}\) may be treated by a slight extension of this theorem, if so desired. Two examples may illuminate the power of this theorem. (1) For any variety \({\mathcal V}\) of bands, \(\underline {\mathcal V}={\mathcal T}\), the trivial variety. Th…
Mathematics Subject Classification ID: 20E34
Mathematics Subject Classification ID: 20E22
Mathematics Subject Classification ID: 20D40
Mathematics Subject Classification ID: 20E36
Mathematics Subject Classification ID: 16D70
zbMATH DE Number: 4040170
zbMATH Keywords: sets of operators
zbMATH Keywords: skew rings
zbMATH Keywords: direct decompositions of groups
zbMATH Keywords: Maschke's problem
zbMATH Keywords: Krull-Schmidt theorem
zbMATH Keywords: finite group
zbMATH Keywords: normal subgroups
zbMATH Keywords: invariant complement
zbMATH Keywords: lattice of band varieties
zbMATH Keywords: orthogroup varieties
zbMATH Keywords: lattice of group varieties
reviewed by: Peter R. Jones
MaRDI profile type: Publication
full work available at URL: 0001-8708(88)90005-9
OpenAlex ID: W1990099391
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description: scientific article; zbMATH DE number 4040170