On two classes of generalised finite T-groups (Q6042203)

A finite soluble group \(G\) is a \(\mathsf{T}\)-group if every subnormal subgroup of \(G\) is normal. These groups have been classified by \textit{W. Gaschütz} [J. Reine Angew. Math. 198, 87--92 (1957; Zbl 0077.25003)] and by \textit{D. J. S. Robinson} [Proc. Am. Math. Soc. 19, 933--937 (1968; Zbl 0159.31002)]. Let \(\sigma=\{ \sigma_{i} \mid i \in I \}\) be a partition of the set of all prime numbers. A subgroup \(X\) of a finite group \(G\) is called \(\sigma\)-subnormal in \(G\) if there is a chain of subgroups \(X=X_{0} \leq X_{1} \leq \dots \leq X_{n}=G\) where for every \(j = 1, \ldots, n\) the subgroup \(X_{j-1}\) is normal in \(X_{j}\) or \(X_{j}/\mathrm{Core}_{X_{j}}(X_{j-1})\) is a \(\sigma_{i}\)-group for some \(i \in I\). A \(\mathsf{T}_{\sigma}\)-group is a finite group in which every \(\sigma\)-subnormal subgroup is normal. The aim of this paper is to study the class of \(\mathsf{T}_{\sigma}\)-groups and other classes which are related to it.