Almost Isomorphic Torsion-Free Abelian Groups and Similarity of Homogeneously Decomposable Groups
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Almost Isomorphic Torsion-Free Abelian Groups and Similarity of Homogeneously Decomposable Groups S. YA. GRINSHPON1 , I. E. GRINSHPON2 and A. I. SHERSTNEVA3
1 Tomsk State University, 36 Lenin Prosp., 634050 Tomsk, Russia. e-mail: [email protected] 2 Tomsk University of Control Systems and Radio Electronics, Tomsk, Russia 3 Tomsk Polytechnic University, Tomsk, Russia
Abstract. In the paper the concept of the similarity of homogeneously decomposable groups is introduced. One investigates the cases in which almost isomorphic groups are similar or isomorphic. Mathematics Subject Classification (2000): 20K99. Key words: almost isomorphic groups, correct group, characteristic, type, homogeneous group, completely decomposable group, fully invariant subgroup.
The well-known set-theoretical Cantor–Schroeder–Bernstein theorem is a source of similar problems in various branches of mathematics including the theory of Abelian groups. Two Abelian groups each of which is isomorphic to a subgroup of the other group, are called almost isomorphic [11]. Two Abelian groups are called almost isomorphic by subgroups with a certain property if each of them is isomorphic to a subgroup of the other group possessing that property. It is natural to pose a question whether almost isomorphic groups are isomorphic. This problem has attracted the attention of many algebraists. The question about isomorphism of Abelian groups which are almost isomorphic by their direct summands is raised in one of Kaplansky’s test problems [12]. For countable reduced p-groups, the answer is positive [12]. However, P. Crawly presented an example of nonisomorphic p-groups each of which is isomorphic to a direct summand of the other group [2]. For torsion-free groups, such examples were constructed by A. Corner and E. Sasiada [1, 17]. In [4], for the class of separable groups, which play an important part in the theory of Abelian groups, L. Fuchs puts the following problem (Problem 28): are two separable torsion-free groups each of which is isomorphic to a pure subgroup of the other group isomorphic? Studying completely decomposable torsion-free groups, which form an important subclass of separable torsion-free groups, de Groot established that completely decomposable torsion-free groups almost isomorphic by pure subgroups are isomorphic [3].
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However, A. P. Mishina and L. A. Skornyakov demonstrated that this is not true. An example of nonisomorphic completely decomposable torsion-free infinite rank groups almost isomorphic by pure subgroups was constructed in [13]. In some papers (for instance, [16, 15, 14, 8, 6, 7]), one studies conditions under which Abelian groups almost isomorphic by pure or fully invariant subgroups are isomorphic. In generalizing the set-theoretical Cantor–Schroeder–Bernstein theorem to Abelian groups, one can conveniently use an approach in which one of the groups is fixed and the other runs over the whole class of Abelian groups. A group A is said to be correct (correct in a class of group
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