Almost isomorphism of Abelian groups and determinability of Abelian groups by their subgroups
Fundamentalʹnaâ i prikladnaâ matematika, Tome 9 (2003) no. 3, pp. 21-36
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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 direct sums of cyclic groups by their subgroups and their proper subgroups, as well as a criterion of correctness of such groups, are obtained.
@article{FPM_2003_9_3_a2,
author = {S. Ya. Grinshpon and A. K. Mordovskoi},
title = {Almost isomorphism of {Abelian} groups and determinability of {Abelian} groups by their subgroups},
journal = {Fundamentalʹna\^a i prikladna\^a matematika},
pages = {21--36},
publisher = {mathdoc},
volume = {9},
number = {3},
year = {2003},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/FPM_2003_9_3_a2/}
}
TY - JOUR AU - S. Ya. Grinshpon AU - A. K. Mordovskoi TI - Almost isomorphism of Abelian groups and determinability of Abelian groups by their subgroups JO - Fundamentalʹnaâ i prikladnaâ matematika PY - 2003 SP - 21 EP - 36 VL - 9 IS - 3 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/FPM_2003_9_3_a2/ LA - ru ID - FPM_2003_9_3_a2 ER -
%0 Journal Article %A S. Ya. Grinshpon %A A. K. Mordovskoi %T Almost isomorphism of Abelian groups and determinability of Abelian groups by their subgroups %J Fundamentalʹnaâ i prikladnaâ matematika %D 2003 %P 21-36 %V 9 %N 3 %I mathdoc %U http://geodesic.mathdoc.fr/item/FPM_2003_9_3_a2/ %G ru %F FPM_2003_9_3_a2
S. Ya. Grinshpon; A. K. Mordovskoi. Almost isomorphism of Abelian groups and determinability of Abelian groups by their subgroups. Fundamentalʹnaâ i prikladnaâ matematika, Tome 9 (2003) no. 3, pp. 21-36. http://geodesic.mathdoc.fr/item/FPM_2003_9_3_a2/