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Equivalent Abelian Groups

Published online by Cambridge University Press:  20 November 2018

J. De Groot
J. De Groot*
Affiliation:
Mathematisch Instituut University of Amsterdam
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Throughout this note all groups are abelian, written additively. We refer to Kurosh (8; 9) for notation, terminology and theorems used without reference. We recall the notion of a serving subgroup (or pure subgroup) of a group . This is a subgroup in which for every natural number n every equation nx = s, s ∊ can be solved provided that it can be solved in . If is torsion-free, “linearly closed” subgroups coincide with serving subgroups and is a serving subgroup if and only if / is torsionfree.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 9 , 1957 , pp. 291 - 297
Copyright
Copyright © Canadian Mathematical Society 1957

References

1. Alexandroff, P. and Hopf, H., Topologie I (Berlin, 1935).Google Scholar
2. Baer, R., Abelian groups without elements of finite order, Duke Math. J., 3 (1937), 68–122.Google Scholar
3. P. Erdös, , On direct decompositions of torsion-free abelian groups, Publ. Math. Debrecen, 3 (1955), 281–288.Google Scholar
4. Fuchs, L., Kertész, A. and Szele, T., Abelian groups in which every serving subgroup is a direct summand, Publ. Math. Debrecen, 3 (1955), 95–105.Google Scholar
5. de Groot, J., An isomorphism criterion for completely decomposable abelian groups, to be published in Math. Ann.Google Scholar
6. Kaplansky, I., Infinite abelian groups (Ann Arbor, 1954).Google Scholar
7. Kulikov, L. Y., On direct decompositions of groups, Ukrain. Mat. Ž., 4 (1952), 230–275, 347–372.Google Scholar
8. Kurosh, A. G., Gruppentheorie (Berlin, 1953).Google Scholar
9. Kurosh, A. G., Theory of groups I (New York, 1955).Google Scholar
10. Levi, F., Abelsche Gruppen mit abzàhlbaren Elementen, Thesis (Leipzig, 1919).Google Scholar
11. Massey, W. S., Some problems in algebraic topology and the theory of fibre bundles, Ann. Math., 62 (1955), 327–359.Google Scholar
12. Pontrjagin, L., The theory of topological commutative groups, Ann. Math., 35 (1934), 361–388.Google Scholar
13. Yang, C. T., On Borsuk's problem, Summa Bras. Math., 3 (1952), 13–20.Google Scholar