Geodesic
A problem on Abelian groups
Sbornik. Mathematics, Tome 34 (1978) no. 4, pp. 461-474 
A. V. Ivanov. A problem on Abelian groups. Sbornik. Mathematics, Tome 34 (1978) no. 4, pp. 461-474. http://geodesic.mathdoc.fr/item/SM_1978_34_4_a3/
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Voir la notice de l'article provenant de la source Math-Net.Ru

We solve Problem 44 in the book by L. Fuchs “Infinite Abelian Groups”, Vol. I, which asks for a classification of the groups $G$ having the following property: if $G$ is contained in the direct sum of reduced groups, then $nG$ for some $n>0$ is contained in a finite direct sum of these groups. A group has this property if and only if it has no unbounded factor groups that are direct sums of periodic cyclic groups. We also consider a generalization of this problem, when instead of the class of all reduced groups we take an arbitrary class of groups. We derive a number of properties of such groups. Bibliography: 8 titles.

[1] L. Fuks, Beskonechnye abelevy gruppy, t. 1, izd-vo “Mir”, Moskva, 1974

[2] A. P. Mishina, L. A. Skornyakov, Abelevy gruppy i moduli, izd-vo “Nauka”, Moskva, 1969 | MR

[3] A. G. Kurosh, “Radikaly v teorii grupp”, Sib. matem. zh., 3:6 (1962), 912–931 | Zbl

[4] V. S. Rokhlina, “Nekotorye klassy abelevykh grupp”, Matem. sb., 83(125) (1970), 214–221 | Zbl

[5] S. Chase, “On direct sums and products of modules”, Pacific J. Math., 12:3 (1962), 847–854 | MR | Zbl

[6] D. Arnold, C. Murley, “Abelian groups $A$, such that $\operatorname{Ham}(A,-)$ preserves direct sums of copies of $A$”, Pacific. J. Math., 56:1 (1975), 7–20 | MR | Zbl

[7] R. Nunke, “Slender groups”, Acta scient. Math., 23:1–2 (1962), 67–73 | MR | Zbl

[8] L. Fuchs, Abelian groups, Publishing House of the Hungarian Academy of Science, Budapest, 1958 | MR