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On groups satisfying the maximal condition on non-normal subgroups. (English) Zbl 0751.20022
The study of groups with different finiteness conditions (in particular maximal conditions) is an old and important part of group theory. This article examines groups with maximal condition on non-normal subgroups – $$\text{Max-}\bar n$$. The main conclusion of the article is the following Theorem. A locally graded group $$G$$ satisfies $$\text{Max-}\bar n$$ if and only if it is of one of the following types: (1) $$G$$ satisfies Max; (2) $$G$$ is a Dedekind group; (3) $$G$$ is a central extension of a Prüfer group by a finitely generated Dedekind group; (4) $$G$$ is the direct product of $$\mathbb{Q}_ 2$$ and a finite Hamiltonian group. Note that condition $$\text{Max-}\bar n$$ for locally almost soluble groups was discussed in an article by the reviewer, N. F. Kuzennyj, N. N. Semko [Dokl. Akad. Nauk Ukr. SSR, Ser. A 1987, No. 1, 9-11 (1987; Zbl 0615.20017)].

##### MSC:
 2e+16 Chains and lattices of subgroups, subnormal subgroups 2e+35 General structure theorems for groups