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Groups with all proper subgroups soluble-by-finite rank. (English) Zbl 1083.20034
Many authors have investigated the structure of groups whose proper subgroups have a given property. In the paper under review groups in which every proper subgroup \(H\) contains a soluble normal subgroup \(K\) such that \(H/K\) has finite Prüfer rank are considered, and the authors prove in particular that a locally (soluble-by-finite) group \(G\) with this property satisfies one of the following conditions: (a) \(G\) contains a soluble normal subgroup \(N\) such that \(G/N\) has finite Prüfer rank; (b) \(G\) is locally soluble; (c) \(G\) contains a soluble normal subgroup \(N\) such that \(G/N\) is isomorphic either to \(\text{PSL}(2,F)\) or to \(\text{Sz}(F)\), where \(F\) is an infinite locally finite field with no infinite proper subfields.

MSC:
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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