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A torsionfree abelian group is called a Butler group if for any torsion group . It has been shown in [DHR] that under any countable pure subgroup of a Butler group of cardinality not exceeding is again Butler. The purpose of this note is to show that this property has any Butler group which can be expressed as a smooth union of pure subgroups having countable typesets.
It is proved that if is an abelian -group with a pure subgroup so that is at most countable and is either -totally projective or -summable, then is either -totally projective or -summable as well. Moreover, if in addition is nice in , then being either strongly -totally projective or strongly -summable implies that so is . This generalizes a classical result of Wallace (J. Algebra, 1971) for totally projective -groups as well as continues our recent investigations in (Arch....
It is well-known that every bounded Abelian group is a direct sum of finite cyclic subgroups. We characterize those non-trivial bounded subgroups of an infinite Abelian group , for which there is an infinite subgroup of containing such that has a special decomposition into a direct sum which takes into account the properties of , and which induces a natural decomposition of into a direct sum of finite subgroups.
Suppose is a subgroup of the reduced abelian -group . The following two dual results are proved: If is countable and is an almost totally projective group, then is an almost totally projective group. If is countable and nice in such that is an almost totally projective group, then is an almost totally projective group. These results somewhat strengthen theorems due to Wallace (J. Algebra, 1971) and Hill (Comment. Math. Univ. Carol., 1995), respectively.
From a paper by A. Angelakis and P. Stevenhagen on the determination of a family of imaginary quadratic fields having isomorphic absolute Abelian Galois groups , we study any such issue for arbitrary number fields . We show that this kind of property is probably not easily generalizable, apart from imaginary quadratic fields, because of some -adic obstructions coming from the global units of . By restriction to the -Sylow subgroups of and assuming the Leopoldt conjecture we show that the...
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