ABSTRACT
The maximal condition was introduced in Chapter 2 and in Chapter 3 it is the turn of the minimal condition. The basic properties of groups with the minimal condition on subgroups are given. A. Yu. Ol'shanskii's groups satisfy both the maximal and minimal condition but for the minimal condition the important role of Chernikov groups is described including the important result that a locally finite group with the minimal condition must be Chernikov. Some of the main results for other minimal conditions, such as the minimal condition on abelian subgroups and on non-abelian subgroups, are given. An extensive discussion is given concerning the minimal condition on normal subgroups and the relationship with Artinian modules over certain group rings is explored. Minimax groups are included in this chapter as a natural way of combining the minimal and maximal conditions and this leads rather naturally to the work that has been done concerning the various weak minimal conditions. The chapter ends with numerous results given for groups with the weak maximal condition.
