A torsionfree abelian group $B$ is called a Butler group if $Bext(B,T)=0$ for any torsion group $T$. It has been shown in [DHR] that under $CH$ any countable pure subgroup of a Butler group of cardinality not exceeding ${\aleph }_{\omega }$ 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 ${\cup }_{\alpha <\mu }{B}_{\alpha }$ of pure subgroups $B_{\alpha }$ having countable typesets.

It is proved that if $A$ is an abelian $p$-group with a pure subgroup $G$ so that $A/G$ is at most countable and $G$ is either $p^{\omega +n}$-totally projective or $p^{\omega +n}$-summable, then $A$ is either $p^{\omega +n}$-totally projective or $p^{\omega +n}$-summable as well. Moreover, if in addition $G$ is nice in $A$, then $G$ being either strongly $p^{\omega +n}$-totally projective or strongly $p^{\omega +n}$-summable implies that so is $A$. This generalizes a classical result of Wallace (J. Algebra, 1971) for totally projective $p$-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 $H$ of an infinite Abelian group $G$, for which there is an infinite subgroup $G_0$ of $G$ containing $H$ such that $G_0$ has a special decomposition into a direct sum which takes into account the properties of $G$, and which induces a natural decomposition of $H$ into a direct sum of finite subgroups.

Suppose $G$ is a subgroup of the reduced abelian $p$-group $A$. The following two dual results are proved: $(*)$ If $A/G$ is countable and $G$ is an almost totally projective group, then $A$ is an almost totally projective group. $(**)$ If $G$ is countable and nice in $A$ such that $A/G$ is an almost totally projective group, then $A$ 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 $K$ having isomorphic absolute Abelian Galois groups $A_K$, we study any such issue for arbitrary number fields $K$. We show that this kind of property is probably not easily generalizable, apart from imaginary quadratic fields, because of some $p$-adic obstructions coming from the global units of $K$. By restriction to the $p$-Sylow subgroups of $A_K$ and assuming the Leopoldt conjecture we show that the...