Correctness of Abelian torsion-free groups and determinability of Abelian groups by their subgroups

S. Ya. Grinshpon   ; A. K. Mordovskoi

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 5 (2014), pp. 16-29

Résumé

An Abelian group $A$ is called correct if for any Abelian group $B$ isomorphisms $A\cong B'$ and $B\cong A'$, where $A'$ and $B'$ are subgroups of the groups $A$ and $B$, respectively, imply the isomorphism $A\cong B$. 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 $A\cong B$. 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 divisible torsion-free groups and completely decomposable groups by their subgroups and their proper subgroups, as well as a criterion of correctness of such groups, are obtained.

Détail

Keywords: almost isomorphism, correctness of Abelian groups, determinability of Abelian groups by their subgroups (their proper subgroups).
Mots-clés : $s$-isomorphism, $t$-isomorphism

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