On generalized Dedekind groups.
Let \(G\) be a finite group. The norm of the group \(G\) consists of the elements normalizing all subgroups, and has been the subject of research for a long time, and recently, for instance, by \textit{M. Herzog, P. Longobardi, M. Maj, A. Mann} [J. Algebra 226, No. 2, 690-713 (2000; Zbl 0959.20031)] and by the authors [Algebra Colloq. 14, No. 4, 605-612 (2007; Zbl 1149.20016)]. The group \(G\) is called an N-group if any of its elements either generates a normal subgroup or is in the norm. In the present paper the authors characterize N-groups, namely, these are exactly nilpotent groups with Sylow-subgroups being N-groups, and at most one of them is not a Dedekind (Hamiltonian) group. Moreover, N-groups of prime power order are also characterized, and as a consequence, one also obtains the characterization of groups in which any element of prime power order either generates a normal subgroup or is in the norm.
scientific article; zbMATH DE number 5651928