Products of locally cyclic groups. (Italian. English summary) Zbl 0864.20015
It is known that if the group \(G\) is the product of finitely many pairwise permutable cyclic subgroups, then \(G\) is supersoluble [see H. Heineken and J. C. Lennox, Arch. Math. 41, 498-501 (1983; Zbl 0509.20017)]. Moreover, M. J. Tomkinson [ibid. 47, 107-112 (1986; Zbl 0596.20026)] proved that, if the factors are Chernikov and locally cyclic, then \(G\) is hypercyclic. In this paper the author proves that if the group \(G\) is the product of finitely many pairwise permutable subgroups which are minimax and locally cyclic, then \(G\) has a finite normal series with locally cyclic factors. Moreover, if the factors are not torsion-free non-cyclic groups, then \(G\) is hypercyclic.
Reviewer: S.Franciosi (Napoli)
MSC:
20E15 | Chains and lattices of subgroups, subnormal subgroups |
20E25 | Local properties of groups |
20E22 | Extensions, wreath products, and other compositions of groups |
20D40 | Products of subgroups of abstract finite groups |
20F16 | Solvable groups, supersolvable groups |
20F14 | Derived series, central series, and generalizations for groups |