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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
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