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Groups with the weak maximal condition for non-subnormal subgroups. (English) Zbl 0928.20025
Summary: A group $$G$$ satisfies max-$$\infty$$-$$\overline{\text{sn}}$$ (the weak maximal condition for non-subnormal subgroups) if there is no infinite ascending chain of non-subnormal subgroups of $$G$$ with all successive indices infinite. Obvious examples of groups $$G$$ with this property are soluble-by-finite minimax groups and groups with all subgroups subnormal. Suppose that $$G$$ belongs to neither of these classes but satisfies max-$$\infty$$-$$\overline{\text{sn}}$$. A structure theorem is established for such groups $$G$$ in the case where $$G$$ is soluble-by-finite, and this is generalized to cover the case of groups $$G$$ that have an ascending series of normal subgroups whose factors are locally soluble-by-finite. Also discussed are groups with max-$$\infty$$-$$\overline{\text{sn}}$$ that are locally finite or locally nilpotent.

##### MSC:
 20E15 Chains and lattices of subgroups, subnormal subgroups 20F19 Generalizations of solvable and nilpotent groups 20E07 Subgroup theorems; subgroup growth 20E34 General structure theorems for groups 20E25 Local properties of groups
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