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On groups with many pronormal subgroups. (English) Zbl 0807.20033
A subgroup $$H$$ of a group $$G$$ is pronormal-sensitive if the set of pronormal subgroups of $$H$$ coincides with the set $$\{H \cap X : X$$ is pronormal in $$G$$}. The main result here is that a finitely generated soluble group in which every subnormal subgroup is pronormal-sensitive is a $$T$$-group (and hence is either abelian or finite). A similar result is obtained for soluble $$FC$$-groups.
##### MSC:
 20F16 Solvable groups, supersolvable groups 20F24 FC-groups and their generalizations 20E15 Chains and lattices of subgroups, subnormal subgroups