Large soluble groups and the control of embedding properties. (Q480375)

A property of subgroups of a group \(G\) is controlled by a class \(\mathfrak C\) of subgroups if all subgroups of \(G\) have this property whenever all subgroups in \(\mathfrak C\) have it. The authors give a survey on results of this form for embedding properties. For instance, for the embedding property of being normal the result of Evans and Kim is mentioned: If \(G\) is locally soluble of infinite rank and all subgroups of infinite rank are normal, then \(G\) is a Dedekind group. Results in the final section are on control of properties of subgroups themselves: Let (just for this review) \(\mathfrak X\) denote the hull of the class of locally graded groups first considered by \textit{N. S. Chernikov} [Ukr. Math. J. 42, No. 7, 855-861 (1990); translation from Ukr. Mat. Zh. 42, No. 7, 962-970 (1990; Zbl 0751.20030)]. If all countable subgroups of infinite rank of the \(\mathfrak X\)-group of infinite rank \(G\) are nilpotent (soluble), then \(G\) itself is nilpotent (soluble) (Theorem 8 (9)).