Supersoluble Groups

We shall term a group G supersoluble if every homomorphic image H9*l of G contains a cyclic normal subgroup different from 1. Supersoluble groups with maximum condition, in particular finite supersoluble groups, have been investigated by various authors: Hirsch, Ore, Zappa and more recently Huppert and Wielandt. In the present note we want to establish the close connection between supersoluble groups and upper nilpotent groups; and we shall use these results to show that the maximum condition is satisfied by the subgroups of every finitely generated supersoluble group. Finally we give a characterization of the groups with maximum condition containing an upper nilpotent subgroup of finite index, since finitely generated supersoluble groups belong in this class. Notations. Z(G) is the center of G and [G, G] is the commutator subgroup of G. An element is said to be of order 0 if it generates an infinite cyclic group.