On direct decompositions of infinite abelian groups with operators
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.
scientific article; zbMATH DE number 4173265