Groups with dense normal subgroups (Q5906570)

This paper is concerned with groups \(G\) in which the set of all normal subgroups is dense in the partially ordered set of subnormal subgroups of \(G\), i.e. if \(H < K\), with \(H\) and \(K\) subnormal in \(G\), and if \(H\) is not maximal subnormal in \(K\), then there exists \(L \vartriangleleft G\) such that \(H < L < K\). Groups with this property are called D-groups. It is clear that D-groups are a generalization of T-groups, i.e. groups in which normality is transitive. The authors present a theory of D-groups, especially soluble D-groups, which is quite analogous to that of soluble T-groups, although there are some important differences. Thus a soluble D-group has derived length \(\leq 3\), but, unlike soluble T-groups, need not be metabelian. Soluble non-abelian D-groups \(G\) are studied under three headings: (i) \(G\) is periodic; (ii) the Fitting subgroup of \(G\) is non-periodic; (iii) \(G\) is non-periodic but the Fitting subgroup is periodic. Detailed information is given on the structure of the groups of each type.