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On hypercentral groups. (English)
Cent. Eur. J. Math. 5, No. 3, 596-606 (2007).
It is a classical result due, in separate papers, to Chernikov and Baumslag [see, for example, {\it D. J. S. Robinson}, Finiteness conditions and generalized soluble groups, Springer-Verlag, Berlin (1972; Zbl 0243.20032 and Zbl 0243.20033)] that if $G$ is a hypercentral group and $G/G’$ is divisible then $G$ is itself divisible. More recently {\it L. Heng, Z. Duan} and {\it G. Chen} [Commun. Algebra 34, No. 5, 1803-1810 (2006; Zbl 1105.20030)] proved that if $G$ is a hypercentral $p$-group, for some prime $p$, then $G$ is divisible-by-finite if and only if $G/G^p$ is finite. In the paper under review the author takes up this theme, first by generalizing the above-mentioned result and second by extending the work of Chernikov and Baumslag. In particular, the author proves: Let $G$ be a hypercentral group. Then $G$ is divisible-by-finite if and only if $G/G’$ is divisible-by-finite. Furthermore, in the case when $G$ is torsion-free and $G/G’$ is divisible-by-finite, then $G$ is divisible. The author also obtains a “$π$-version” of this result, where $π$ is a set of primes. He also proves that if $G$ is hypercentral and $G/G’$ is actually finite then $G$ is a periodic divisible-by-finite group of central height less than $ω2$ but not $ω$.
Reviewer: Martyn Dixon (Tuscaloosa)
Classification: H45
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