Groups with maximum conditions

Abstract

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.