Illinois Journal of Mathematics

Pure subgroups of completely decomposable groups and a group class problem

Daniel Herden and Lutz Strüngmann

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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 $G^{(\kappa)}$?

Article information

Source
Illinois J. Math., Volume 55, Number 4 (2011), 1533-1549.

Dates
First available in Project Euclid: 12 July 2013

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1373636695

Digital Object Identifier
doi:10.1215/ijm/1373636695

Mathematical Reviews number (MathSciNet)
MR3082880

Zentralblatt MATH identifier
1280.20059

Subjects
Primary: 20K15: Torsion-free groups, finite rank 20K20: Torsion-free groups, infinite rank 20K25: Direct sums, direct products, etc. 15A36

Citation

Herden, Daniel; Strüngmann, Lutz. Pure subgroups of completely decomposable groups and a group class problem. Illinois J. Math. 55 (2011), no. 4, 1533--1549. doi:10.1215/ijm/1373636695. https://projecteuclid.org/euclid.ijm/1373636695


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