It still seems to be unknown
whether there exist Noetherian groups ( = groups with maximum condition on
subgroups) that are not almost polycyclic, i.e., possess a soluble normal subgroup of
finite index. However, the existence of even finitely generated infinite simple groups
shows that in general a group whose subnormal subgroups satisfy the maximum
condition need not be almost polycyclic. The following theorem gives a number of
criteria for a group satisfying a weak form of the maximum condition to be almost
polycyclic.
