Abstract
An Abelian group A is called correct if for any Abelian group B isomorphisms A ≅ B′ and B ≅ A′, where A′ and B′ are subgroups of the groups A and B, respectively, imply the isomorphism A ≅ B. We say that a group A is determined by its subgroups (its proper subgroups) if for any group B the existence of a bijection between the sets of all subgroups (all proper subgroups) of groups A and B such that corresponding subgroups are isomorphic implies A ≅ B. In this paper, connections between the correctness of Abelian groups and their determinability by their subgroups (their proper subgroups) are established. Certain criteria of determinability of direct sums of cyclic groups by their subgroups and their proper subgroups, as well as a criterion of correctness of such groups, are obtained.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
K. Borsuk, Theory of Retracts [in Russian], Mir, Moscow (1971).
R. Bumby, “Modules which isomorphic to submodules of each other,” Arch. Math., 16, 184–185 (1965).
I. Cornel, “Some ring theoretical Schroeder—Bernstein theorems,” Trans. Amer. Math. Soc., 132, 335–351 (1968).
P. Crawly, “Solution of Kaplansky’s test problem for primary Abelian groups,” J. Algebra, No. 4, 413–431 (1965).
P. Eklof and G. Sabbagh, “Model-completions and modules,” Ann. Math. Logic, 2, 251–299 (1971).
L. Fuchs, Infinite Abelian Groups, Vol. 1, Academic Press (1970).
S. Ya. Grinshpon, “f.i.-Correctness of torsion free Abelian groups,” Abelian Groups and Modules, No. 8, 65–79 (1989).
S. Ya. Grinshpon, “f.i.-Correct Abelian groups,” Uspekhi Mat. Nauk, No. 6, 155–156 (1999).
J. de Groot, “Equivalent Abelian groups,” Canad. J. Math., No. 9, 291–297 (1957).
R. Holzsager and C. Hallahan, “Mutual direct summands,” Arch. Math., 25, 591–592 (1974).
B. Jonson, “On direct decomposition of torsion free Abelian groups,” Math. Scand., No. 2, 361–371 (1959).
I. Kaplansky, Infinite Abelian Groups, Univ. of Michigan Press, Michigan (1954).
I. A. Prikhod’ko, “E-correct Abelian groups,” Abelian Groups and Modules, 90–99 (1984).
S. K. Rososhek, “Purely correct modules,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 10, 143–150 (1978).
S. K. Rososhek, “Strictly purely correct torsion free Abelian groups,” Abelian Groups and Modules, 143–150 (1979).
A. I. Sherstneva, “U-sequences and almost isomorphism of Abelian p-groups by fully invariant subgroups,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 5, 72–80 (2001).
V. Trnkova and V. Koubek, “The Cantor-Bernstein theorem for functors,” Comment. Math. Univ. Carolin., 14, 197–204 (1973).
Author information
Authors and Affiliations
Additional information
__________
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 3, pp. 21–36, 2003.
Rights and permissions
About this article
Cite this article
Grinshpon, S.Y., Mordovskoi, A.K. Almost isomorphism of Abelian groups and determinability of Abelian groups by their subgroups. J Math Sci 135, 3281–3291 (2006). https://doi.org/10.1007/s10958-006-0158-y
Issue Date:
DOI: https://doi.org/10.1007/s10958-006-0158-y