FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2003, VOLUME 9, NUMBER 3, PAGES 21-36

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

S. Ya. Grinshpon

A. K. Mordovskoi

Abstract

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An Abelian group $A$ is called correct if for
any Abelian group $B$ isomorphisms $A$@
B' and $B$@ A', where $A\text{'}$ and $B\text{'}$ 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.

Location: http://mech.math.msu.su/~fpm/eng/k03/k033/k03303h.htm.

Last modified: January 24, 2005.