Soluble minimal non-(finite-by-Baer)-groups.
Let \(X\) be a class of groups. A group \(G\) is said to be `minimal non-\(X\)' if it does not belong to \(X\) but all of its proper subgroups are \(X\)-groups. Many authors have investigated the structure of minimal non-\(X\)-groups for several choices of the class \(X\). In particular, M. Xu in 1966 proved that an infinite soluble group \(G\) is minimal non-Baer if and only if \(G\) contains a normal subgroup \(N\) such that \(G/N\) is an infinitely generated minimal non-nilpotent group having a maximal subgroup (i.e., \(G/N\) is an \(AN^*\)-group, and hence it is a metabelian Chernikov \(p\)-group, for some prime \(p\)), \(\langle x\rangle N\) is a soluble Baer group for all \(x\in G\) and \(HN<G\) for every proper subgroup \(H\) of \(G\). (Here a `Baer group' is a group in which every cyclic subgroup is subnormal.) In the article under review the authors give the following characterization of \textit{minimal non-(finite-by-Baer)-groups}. Theorem. Let \(G\) be a soluble minimal non-(finite-by-Baer)-group. Then for all integers \(n\geq 2\) the following properties hold: 1) \(G/\gamma_n(G')\) is a minimal non-(finite-by-Abelian)-group; 2) \(\langle x,y\rangle\gamma_n(G')\) is a soluble finite-by-Baer-group for all \(x,y\in G\); 3) If \(G=H\gamma_n(G')\), then \(G=H\) for all subgroups \(H\) of \(G\). -- Conversely, if \(G\) contains a normal subgroup \(N\) satisfying the above conditions, then \(G\) is a soluble minimal non-(finite-by-Baer)-group.
scientific article; zbMATH DE number 5788849