Noetherian Modules Over Hyperfinite Groups

[摘要] Let G be a group and A a ZG-module. If A = Af ⊕ Af, where Af is a ZG-submodule of A such that each irreducible ZG-factor of Af is finite and the ZG-submodule Af of A has no nonzero finite ZG-factors, then A is said to have an f-decomposition. If G is a hyperfinite locally soluble group, then it is known that any artinian ZG-module A has an f-decomposition. In this thesis, especially by investigating the properties of the torsion-free noetherian ZLG-modules, we prove that any noetherian ZG-module A over a hyperfinite locally soluble group G has an f-decomposition, too. Further, the structure of the noetherian ZG-submodule Af is well described and the structure of the noetherian ZG-submodule 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 ZG-module A, we have that: (1) A has an f-decomposition; (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 ZG-composition 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 ZG-module A over a hyper-(cyclic or finite) locally soluble group G has an f-decomposition. 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 ZG-module, 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 ZG-module with A = Af, then E splits conjugately over A. A number of questions are given at the end of the work.

[发布机构] University:University of Glasgow

[关键词] Mathematics