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Almost isomorphism of Abelian groups and determinability of Abelian groups by their subgroups

Abstract

An Abelian group A is called correct if for any Abelian group B isomorphisms AB′ and BA′, where A′ and B′ are subgroups of the groups A and B, respectively, imply the isomorphism AB. 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 AB. 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.

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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 3, pp. 21–36, 2003.

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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

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Keywords

  • Abelian Group
  • Cyclic Group
  • Proper Subgroup