A generalization of T-groups (Q1864525)

Recall that a `T-group' is a group in which normality is transitive. The author studies the following generalization. A group \(G\) is called a `Hall T-group' if it has a nilpotent normal subgroup \(N\) such that \(G/N'\) is a T-group. (The motivation for the definition is a well-known theorem of P.~Hall: if \(N\) is a nilpotent normal subgroup of \(G\) and \(G/N'\) is nilpotent, then \(G\) is nilpotent). The author obtains results on the structure of soluble Hall T-groups. For example, they are nilpotent-by-Abelian and locally supersoluble; also a finitely generated soluble Hall T-group is either finite or nilpotent. These are reminiscent of results of the reviewer on T-groups [Proc. Camb. Philos. Soc. 60, 21-38 (1964; Zbl 0123.24901)]. The main result of the paper is: Theorem 3.6. Let \(G=HK\) where \(H\) is a soluble normal Hall T-subgroup and \(K\) is a subnormal locally supersoluble subgroup. Then \(G\) is locally supersoluble.