© Hindawi Publishing Corp. ON MINIMAL ARTINIAN MODULES AND MINIMAL ARTINIAN LINEAR GROUPS

Abstract. The paper is devoted to the study of some important types of minimal artinian linear groups. The authors prove that in such classes of groups as hypercentral groups (so also, nilpotent and abelian groups) and FC-groups, minimal artinian linear groups have precisely the same structure as the corresponding irreducible linear groups. 2000 Mathematics Subject Classification. 20E36, 20F28. Let F be a field, A a vector space over F. The group GL(F,A) of all automorphisms of A and its distinct subgroups are the oldest subjects of investigation in Group The-ory. For the case when A has a finite dimension over F, every element of GL(F,A) defines some nonsingular n×n-matrix over F, where n = dimF A. Thus, for the finite-dimensional case, the theory of linear groups is exactly the theory of matrix groups. That is why the theory of finite-dimensional linear groups is one of the best developed in algebra. However, for the case when dimF A is infinite, the situation is totally dif-ferent. The study of this case always requires some essential additional restrictions. Thus, the transition from the study of finite groups to the study of infinite groups generated the finiteness conditions. It is natural to apply these finiteness condition