Minimal non-\(\mathcal F\)-groups. (Q1042612)

Let \(\mathcal F\) be a subgroup closed saturated formation of groups. \textit{V. N. Semenchuk} [Algebra Logika 18, 348-382 (1979; Zbl 0463.20018); translation in Algebra Logic 18, 214-233 (1980)] has described the minimal non-\(\mathcal F\)-groups. The present paper describes in great detail the minimal non-\(\mathcal F\)-groups, where \(\mathcal F\) is the class of extensions of Abelian groups by groups from a saturated subgroup closed formation (Theorem 2). \textit{I. D. Macdonald} [Math. Z. 76, 270-282 (1961; Zbl 0104.02202)] has characterised those groups whose three-generator subgroups are metabelian. The authors obtain the following result: If \(G\) is a finite soluble non-metabelian group all of whose proper subgroups are metabelian, then either \(G\) can be generated by 3 elements or \(G\) is a 2-group (Theorem 3).