The authors consider \(W\)-groups with the above property, here \(G\) is a \(W\)-group iff every finitely generated non-nilpotent subgroup of \(G\) has a finite non-nilpotent homomorphic image. Further \(G\) has finitely many maximal normalizers (has a finite normalizing set) of \(\theta\)-subgroups iff the normalizer of every \(\theta\)-subgroup is contained in one of these normalizers (in one of the proper subgroups belonging to this set).
The following is shown (Theorem A): If \(G\) is a \(W\)-group with finitely many maximal normalizers of non-Abelian subgroups, then \(G'\) is finite. This is the consequence of a more general result (Theorem B): If \(G\) is a \(W\)-group with finite normalizing set for non-Abelian subgroups, then \(G'\) is finite. – Notice that the conditions in Theorem B are inherited by subgroups and homomorphic images, and this is not so for those of Theorem A. – This article generalizes results of De Mari and de Giovanni on finitely many normalizers and of Romalis and Sesekin on groups with all non-Abelian subgroups normal.