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Groups factorized by pairwise permutable abelian subgroups of finite rank. (English) Zbl 1378.20044

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 \(6\).

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

Citations:

Zbl 0509.20017
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