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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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