On products of normal hypercyclic subgroups. (Q1879043)

It is well-known that the product of two normal hypercyclic subgroups need not be hypercyclic, even in finite groups. Many additional conditions have been proposed which will ensure hypercyclicity, and the article under review adds to this body of work. A group \(G\) such that for some \(i\) every subgroup of \(\gamma_i(G)\) is normal in \(G\) is referred to by the acronym ``soging''. Note that this implies hypercyclicity of \(G\). The main result proved~is: Theorem 1.1. Let \(G=HK\) where \(H\triangleleft G\), \(K\triangleleft G\), \(H\) is soging and \(K\) is hypercyclic. Then \(G\) is hypercyclic. Recall that a \(T\)-group is a group in which normality is transitive. It is well-known that soluble \(T\)-groups are metabelian, and it follows that a soluble \(T\)-group is soging. The second main result~is: Theorem 1.2. Let \(G=HK\) where \(H\triangleleft G\) and \(K\) is subnormal in \(G\). Assume that \(H\) is a soluble Hall-\(T\)-group and \(K\) is hypercyclic. Then \(G\) is hypercyclic. Here \(H\) is a Hall-\(T\)-group if it has a nilpotent normal subgroup \(N\) such that \(H/N'\) is a \(T\)-group.