Amberg, Bernhard; Sysak, Yaroslav P. Groups factorized by pairwise permutable abelian subgroups of finite rank. (English) Zbl 1378.20044 Adv. Group Theory Appl. 2, 13-24 (2016). A famous result of Itô shows that any group factorized by two abelian subgroups is metabelian. Later, H. Heineken and J. C. Lennox [Arch. Math. 41, 498–501 (1983; Zbl 0509.20017)] proved that a group factorized by finitely many pairwise permutable abelian subgroups is polycyclic, provided that all factors are finitely generated. The paper under review deals with the structure of groups factorized by finitely many pairwise permutable abelian subgroups of finite rank. It is proved that such a group must be hyperabelian of finite rank. Moreover, the authors prove that if \(G=ABC\) is a non-periodic group factorized by three pairwise permutable locally cyclic subgroups \(A,B,C\) and \(A\cap B\cap C=\{1\}\), then \(G\) is soluble and it has derived length at most \(4\) and Prüser rank at most \(6\). Reviewer: Francesco de Giovanni (Napoli) Cited in 1 Document MSC: 20E22 Extensions, wreath products, and other compositions of groups 20D40 Products of subgroups of abstract finite groups 20F16 Solvable groups, supersolvable groups 20E07 Subgroup theorems; subgroup growth 20E25 Local properties of groups Keywords:factorized group; finite rank; locally cyclic group Citations:Zbl 0509.20017 PDFBibTeX XMLCite \textit{B. Amberg} and \textit{Y. P. Sysak}, Adv. Group Theory Appl. 2, 13--24 (2016; Zbl 1378.20044) Full Text: DOI