The Abelianization of hypercyclic groups. (Q931580)

It was shown in the literature that the Abelianization of a hypercentral group has a considerable influence on the structure of the group itself. Since hypercentral groups are hypercyclic groups, it is natural to ask whether the results obtained for hypercentral groups extend to hypercyclic groups. In the article under review, different aspects of this question are studied and the more substantial results obtained for a hypercyclic group \(G\) are the following: a) If \(G/G'\) is \(2\)-divisible, then \(G\) is \(2\)-divisible; b) if \(G/G'\) is \(2'\)-divisible, then \(G\) is \(2'\)-divisible; if \(G/G'\) is divisible-by-finite of odd order, then \(G/V\) is divisible-by-finite of odd order, where \(V\) is the intersection of the lower central series of \(O_{2'}(G)\).