On direct decompositions of infinite abelian groups with operators
type: Item
instance of: scholarly article
title: On direct decompositions of infinite abelian groups with operators
zbMATH Open document ID: 0713.20052
DOI: 10.1007/BF01061301
author: D. I. Zajtsev
published in: Ukrainian Mathematical Journal
publication date: 1988-01-01T00:00:00Z
review text: Let A be an abelian group and G a group of automorphisms of A. Then A is a \({\mathbb{Z}}G\)-module. If \(0<B<A\) is a composition series for the \({\mathbb{Z}}G\)-module A with one of B, A/B finite and the other infinite, then as abelian group B has a direct complement D in A, i.e. \(A=B\oplus D\) for some subgroup D. In an earlier paper the author considered the question of whether there exists such a D which is G-invariant, i.e. a \({\mathbb{Z}}G\)-submodule. In the present paper this question is considered under the weaker assumption that A has merely a finite composition series (rather than one of length 2). Theorem 1 gives a canonical direct decomposition of such an A relative to a normal subgroup H of G consisting of elements with finite conjugacy classes (``FC-elements''), whence Theorem 2: If A is a faithful G-module having a finite composition series with just one factor infinite, and if G has infinitely many FC- elements, then A is the direct sum of a finite submodule and an infinite simple submodule. Corollary. A module over a hypercyclic group, having a finite composition series, has a canonical direct decomposition (as defined by Theorem 1). From Theorem 3 it follows that if G is locally supersolvable and A has a composition series \(0<B<A\) with B finite, A/B infinite, then B has a direct G-invariant complement. The paper ends with examples showing the limitations on such results.
Mathematics Subject Classification ID: 20K25
Mathematics Subject Classification ID: 20C07
zbMATH DE Number: 4173265
zbMATH Keywords: abelian group
zbMATH Keywords: group of automorphisms
zbMATH Keywords: composition series
zbMATH Keywords: \({\mathbb{Z}}G\)- module
zbMATH Keywords: direct complement
zbMATH Keywords: direct decomposition
zbMATH Keywords: elements with finite conjugacy classes
zbMATH Keywords: FC-elements
zbMATH Keywords: direct sum
zbMATH Keywords: hypercyclic group
MaRDI profile type: Publication
cites work: Q5654235
cites work: Irreducible groups of automorphisms of abelian groups
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cites work: Maximal subgroups and chief factors of certain generalized soluble groups
description: scientific article; zbMATH DE number 4173265