On induced modules over group rings of groups of finite rank.

Let G be a group and k be a field. A kG-module M is said to be imprimitive if there are a subgroup H < G and a kH-submodule N ≤M such that M = N ⊗kH KG. If the module M is not imprimitive then it is said to be primitive. A representation of the group G is said to be primitive if the module of the representation is primitive. Let G be a group of finite rank r(G) and k be a field. A kG-module M is said to be semiimprimitive if there are subgroup H < G and a kH-submodule N ≤M such that r(H) < r(G) andM = N⊗kHKG. If the module M is not semi-imprimitive then it is said to be semi-primitive. A representation of the group G is said to be semi-primitive if the module of the representation is semi-primitive. An element g ∈ G ( a subgroup H ≤ G) is said to be orbital if |G : CG(g)| <∞ (|G : NG(H)| <∞). The set ∆ (G) of all orbital elements of G is a characteristic subgroup of G which is said to be the FC-center of G. In [1] Harper shoved that any finitely generated not abelian-by-finite nilpotent group has an irreducible primitive representation over any not locally finite field. In [3] we proved that in the class of soluble groups of finite rank with the maximal condition for normal subgroups only polycyclic groups may have irreducible primitive faithful representations over a field of characteristic zero. In [2] Harper proved that if a polycyclic group G has a faithful irreducible semi-primitive representation then A ⋂︀ ∆ (G) ≠ 1 for any orbital subgroup A of G. It is well known that any polycyclic group is liner and has finite rank.