Rank restricting properties of finitely generated soluble groups
If \(H\leq K\) are subgroups of a group G, then the set \(H^{1/k}=\{g\in K|\quad g^ n\in H\quad for\quad some\quad n\geq 1\}\) is called the isolator of H in K. In this paper we study finitely generated soluble groups with the following property: (*) For all pairs of subgroups H, K of the group G, where H is subnormal in K, \(H^{1/k}\) is a subgroup. The main result of the paper shows that such a group is periodic-by- finite (Prüfer) rank. Together with a converse, saying that a finitely generated soluble group of finite (Prüfer) rank contains a subgroup of finite index with property (*) [\textit{A. Rhemtulla}, \textit{B. A. F. Wehrfritz}: Rocky Mt. J. Math. 14, 415-421 (1984; Zbl 0545.20026)], this gives an almost complete characterization of finitely generated soluble groups with (*). Our paper [Can. J. Math. (to appear; Zbl 0544.20034)] plays an important role for the proofs.
scientific article; zbMATH DE number 3871641
scientific article