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Noetherian modules over hyperfinite groups

Let G be a group and A a ZGmodule. If A = Af ⊕ Af, where Af is a ZGsubmodule of A such that each irreducible ZGfactor of Af is finite and the ZGsubmodule Af of A has no nonzero finite ZGfactors, then A is said to have an fdecomposition. If G is a hyperfinite locally soluble group, then it is known that any artinian ZGmodule A has an fdecomposition. In this thesis, especially by investigating the properties of the torsionfree noetherian ZLGmodules, we prove that any noetherian ZGmodule A over a hyperfinite locally soluble group G has an fdecomposition, too. Further, the structure of the noetherian ZGsubmodule Af is well described and the structure of the noetherian ZGsubmodule Af is discussed in detail. If G is a Cernikov group (not necessarily locally soluble) or, more generally, if G is a finite extension of a periodic abelian group with pi(G) < infinity, where pi(G) = {prime p; G has an element of order p} , then, for any noetherian ZGmodule A, we have that: (1) A has an fdecomposition; (2) Af is finitely generated as an abelian group and G/CG(Af) is finite; and (3) Af is torsion as a group and has a finite ZGcomposition series as well as a finite exponent. Moreover, we have generalized Zaicev's results about modules over hyperfinite locally soluble groups to modules over hyper(cyclic or finite) groups. In fact, we have got the following results: Theorem C: Any periodic artinian ZGmodule A over a hyper(cyclic or finite) locally soluble group G has an fdecomposition. Theorem D: Let E be an extension of a periodic abelian group A by a hyper(cyclic or finite) locally soluble group G. If A is an artinian ZGmodule, then E splits conjugately over A modulo Af. And Theorem E: Let E be an extension of an abelian group A by a hyper(cyclic or finite) locally soluble group G. If A is a noetherian ZGmodule with A = Af, then E splits conjugately over A. A number of questions are given at the end of the work.
