## Illinois Journal of Mathematics

### 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 $G^{(\kappa)}$?

#### 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

#### 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

#### References

• D. M. Arnold, Notes on Butler groups and balanced extensions, Boll. Unione Mat. Ital. A (6) 5 (1986), 175–184.
• D. M. Arnold, Abelian groups and representations of finite partially ordered sets, CMS Books in Mathematics, Springer-Verlag, New York, 2000.
• D. M. Arnold and K. M. Rangaswamy, A note on countable Butler groups, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 10 (2007), 605–611.
• L. Bican and L. Salce, Butler groups of infinite rank, Abelian group theory, Lecture Notes in Math., vol. 1006, Springer-Verlag, New York, 1965, pp. 680–698.
• M. C. R. Butler, A class of torsion-free abelian groups of finite rank, Proc. Lond. Math. Soc. 15 (1965), 680–698.
• L. Fuchs, Infinite Abelian groups, vols. 1 & 2, Academic Press, New York, 1970 & 1973.
• L. Fuchs, A survey on Butler groups of infinite rank, Contemp. Math. 171 (1994), 121–139.
• M. Gryga, Die Einbettung fast vollständig zerlegbarer Gruppen im globalen Fall, Staatsexamensarbeit, University of Duisburg-Essen, 2009.
• D. Herden and L. Strüngmann, Pure subgroups of completely decomposable groups–-An algorithmic approach, Models, modules and Abelian groups, de Gruyter, Berlin, 2008, pp. 169–186.
• A. Mader, Almost completely decomposable groups, Algebra, logic and applications, vol. 13, Gordon and Breach Science Publishers, Amsterdam, 2000.
• A. Mader and L. Strüngmann, Bounded essential extensions of completely decomposable groups, J. Algebra 229 (2000), 205–233.
• A. Mader and L. Strüngmann, A class of finitely Butler groups and their endomorphism rings, Hokkaido Math. J. 37 (2008), 399–425.
• S. Shelah and L. Strüngmann, It is consistent with ZFC that $B_1$ groups are not $B_2$ groups, Forum Math. 15 (2003), 507–524.
• L. Strüngmann, On endomorphism rings of $B_1$-groups that are not $B_2$-groups when adding $\aleph_4$ reals to the ground model, Proc. Amer. Math. Soc. 137 (2009), 3657–3668.
• C. Tschinke, Die Einbettung fast vollständig zerlegbarer Gruppen im lokalen Fall – Ein Algorithmus und seine Realisierung in Maple, Staatsexamensarbeit, University of Duisburg-Essen, 2009.