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On cancellable abelian groups. (English) Zbl 1407.22002

An abelian group \(N\) equipped with the discrete topology is called cancellable if for any two abelian topological groups \(G\) and \(H\), the product group \(G \times N \cong H \times N\) if and only if \(G \cong H\), where the symbol \(\cong\) means the topological isomorphism the between groups.
In the paper under review the authors show that the additive group \(\mathbb{Z}\) of the integers is cancellable. This answers a problem raised by A. Arhangel’skii and M. Tkachenko in [Topological groups and related structures. Hackensack, NJ: World Scientific; Paris: Atlantis Press (2008; Zbl 1323.22001)]. The authors also show that every finitely generated abelian group is cancellable. Furthermore, they show that a divisible group \(D\) is cancellable if and only if the maximal torsion-free subgroup of \(D\) is the dirct sum of a finite number of copies of the rationals and for each prime \(p\), the \(p\)-primary component of \(D\) is the direct sum of a finite number of copies of the quasi-cyclic group \(\mathbb{Z}(p^\infty)\).

MSC:

22A05 Structure of general topological groups
54H11 Topological groups (topological aspects)

Citations:

Zbl 1323.22001
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References:

[1] Arhangel’skii, A. V.; Tkachenko, M., Topological Groups and Related Structures (2008), World Scientific: World Scientific Singapore · Zbl 1323.22001
[2] Engelking, R., General Topology, Sigma Series in Pure Math, vol. 6 (1989), Heldermann: Heldermann Berlin · Zbl 0684.54001
[3] Fuchs, L., Infinite Abelian Groups, vol. I (1970), Academic Press: Academic Press New York · Zbl 0209.05503
[4] Hofmann, K. H.; Morris, S., The Structure of Compact Groups. A Primer for the Student—A Handbook for the Expert, De Gruyter Studies in Mathematics, vol. 25 (1998), Walter de Gruyter & Co.: Walter de Gruyter & Co. Berlin, xviii+835 pp · Zbl 0919.22001
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