Nilpotent groups with every finite homomorphic image cyclic (Q1096014)

The authors call a group an FHC-group if every finite homomorphic image of it is cyclic, and an AFHC-group if it is, moreover, abelian. Arbitrary FHC-groups can be arbitrarily complicated, as they include all infinite simple groups, and also some ingenious examples made by \textit{G. Baumslag} [J. Aust. Math. Soc. 10, 497-498 (1969; Zbl 0214.274); Bull. Aust. Math. Soc. 6, 313-314 (1972; Zbl 0226.20024)] of one-relator and of locally free FHC-groups. A characterization of AFHC-groups flows from a note by \textit{R. Gilmer} [Am. Math. Mon. 81, 1036-1037 (1974)]. The authors now study nilpotent FHC-groups, which in this review will be called NFHC- groups. Only a selection of the numerous results can be quoted here. If the derived group of an FHC-group is finite, then the group is abelian (or, in the reviewer's notation, \(FHC\cap FD=AFHC)\). If a group is FHC, then the factor group by its derived group is AFHC, and all further lower central factor groups are divisible; and for nilpotent groups the converse is also true. The torsion subgroup of an NFHC-group is AFHC. Examples in the final section of the paper show that the upper central factors of NFHC-groups need not be FHC-groups; in fact they can be quite arbitrary abelian groups. Throughout the paper there is a clear discussion of its relation to known results.