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On the Wielandt subgroup of infinite soluble groups. (English) Zbl 0726.20023

Denote by \(\omega\) (G) the Wielandt subgroup of a group G, that is the intersection of the normalizers of all subnormal subgroups of G. The authors have two main results. If G is finitely generated soluble-by- finite with finite Prüfer rank then \(\omega\) (G) is contained in the FC-centre of G. Also if G is a polycyclic group which is either (a) metanilpotent or (b) abelian-by-finite, then \(\omega\) (G)/Z(G) is finite. There are examples showing the limits of the results.

MSC:

20F16 Solvable groups, supersolvable groups
20E07 Subgroup theorems; subgroup growth
20E15 Chains and lattices of subgroups, subnormal subgroups
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