Pure subgroups of completely decomposable groups and a group class problem

Abstract

In the work of Herden and Strüngmann (In Models, modules and Abelian groups (2008) 169–186 de Gruyter), an embedding problem for torsion-free Abelian groups was considered. It was shown for a large class of such groups, including the class of all bounded extensions of completely decomposable groups, that any member of the class can be purely embedded into some completely decomposable group. Moreover, an algorithm was given that determines explicitly the pure embedding and the completely decomposable overgroup. We continue the approach from the work of Herden and Strüngmann (In Models, modules and Abelian groups (2008) 169–186 de Gruyter) improving the algorithm and extending the main theorem to a broader class of torsion-free Abelian groups including some Hawaiian groups from the article of Mader and Strüngmann (J. Algebra 229 (2000) 205–233) and thus complementing the main result from the article of Strüngmann (Proc. Amer. Math. Soc. 137 (2009) 3657–3668).

A byproduct and starting point for this generalization will be a discussion of the following group class problem: Which groups $G$ have the property that for any cardinal $\kappa$ any subgroup $U$ of the direct sum $G^{(\kappa)}$ is the kernel of some endomorphism of $G^{(\kappa)}$?