On a class of locally supersoluble groups (Q1862688)

It is well known that, even for finite groups, the product of two normal supersoluble subgroups need not be supersoluble. The author gives some extra conditions which ensure that the conclusion is valid. A group \(G\) is called an HCA-group if it has a nilpotent normal subgroup \(N\) such that \(G/N'\) is cyclic-by-Abelian. It is easy to see that HCA-groups are nilpotent-by-(finite Abelian) and hypercyclic, and hence locally supersoluble. The main result of the article is Theorem 3.4. Let \(G=HK\), where \(H\) is a normal HCA-subgroup and \(K\) is a subnormal locally supersoluble subgroup of \(G\). Then \(G\) is locally supersoluble.