Hopfian Algebraically Compact Abelian Groups

Abstract

The subject matter of the paper encompasses Abelian group theory, module theory, and topology. Hopfian groups appeared in combinatorial group theory, which appreciably determined characteristic features of the initial characterization of Hopfian groups. Methods and problems in combinatorial group theory served as tools for describing Hopfian groups and their connections with other classes of groups. In 1921 J. Nielsen published a paper [1] which improves on the research techniques for finitely generated subgroups of free groups developed in his previous works. Nielsen’s results imply an important consequence which says that a finitely generated free group cannot be isomorphic to its proper factor group; i.e., a homomorphic mapping of such a group onto itself always has a trivial kernel. Now this property, named after Heinz Hopf, is referred to as Hopficity. The reason for this is the following. In [2, 3], Hopf explored mappings of one two-dimensional variety into another. These mappings induce homomorphisms of the fundamental group of the first variety onto the fundamental group of its image. Using topological methods, Hopf showed that the groups mentioned cannot be isomorphic to any of their proper factor groups. The simplest case is one in which the fundamental group in question is free and has finite rank. Surprisingly, over a span of thirty years, nobody noticed that Nielsen had already given a purely algebraically proof of the fact that free groups of finite rank are Hopfian—in fact, ten years before Hopf posed his question. Hopfian groups (including Abelian ones) have been extensively studied by many algebraists (for a bibliography, see [4]). To some extent, the present paper can be regarded as a continuation of [4]. For the reader’s convenience, here we lay out some information concerning the notation and ∗Supported by the Russian Ministry of Education and Science, gov. contract No. 14.B37.21.0354.