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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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References:
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