On \(U\)-normal subgroups. (Q1430466)

A subgroup \(A\) of a group \(G\) is abnormal in \(G\) if \(g\in\langle A,A^g\rangle\) for all \(g\in G\). In this paper subgroups which are either normal or abnormal are called \(U\)-normal subgroups, and a group \(G\) is called a \(U\)-group if all subgroups of \(G\) are \(U\)-normal. A group \(G\) is called a \(TU\)-group if \(U\)-normality is a transitive property in the group \(G\) (by analogy with the familiar class of \(T\)-groups in which normality is transitive) and groups in which all subgroups are \(TU\)-groups are called \(\overline TU\)-groups. It is easy to see that every \(TU\)-group is a \(T\)-group, about which numerous papers have been written (and the paper under review lists many of these). The main results are Theorems 2.4, 3.1 and 4.4. Theorem 2.4 gives a classification of the periodic soluble \(TU\)-groups. These essentially fall into four types, but Corollary 2.5 makes the classification yet more precise. Theorems 3.1 and 4.4 treat the non-periodic soluble \(TU\)-groups, in a fashion somewhat akin to the fundamental paper of \textit{D. J. S. Robinson} [Proc. Camb. Philos. Soc. 60, 21-38 (1964; Zbl 0123.24901)] dealing with \(T\)-groups.