Density of indecomposable locally finite groups
Let \(\theta\) be a cardinal number. A group \(G\) is called \(\theta\)-indecomposable if, for every increasing sequence \(\langle G\mid i<\theta\rangle\) of subgroups with union \(G\), there is \(i <\theta\) such that \(G=G_i\). Moreover, a group \(E\) is said to be a locally finite existentially closed group if it contains an isomorphic copy of every finite group and any two isomorphic finite subgroups of \(E\) are conjugate in \(E\) (see the universal groups in [\textit{O. Kegel} and \textit{B. A. F. Wehrfritz}, Locally Finite Groups. New York: Elsevier (1973; Zbl 0259.20001)]). The main result of the paper is the following. Let \(G\) be a locally finite group. Then, under some very weak hypotheses on \(|G|\), there exists a locally finite existentially closed group \(H\) such that \(|H|=|G|\), \(G\leq H\) and such that it is \(\theta\)-indecomposable for any regular cardinal \(\theta\neq cf(|G|)\). We remark that this result is also used in two subsequent works by the same author.
scientific article; zbMATH DE number 7367420
scientific article