Mixed Groups

Abstract

AbstractMixed groups form the most general class of abelian groups, and traditionally, the main question is to find out how the torsion and torsion-free parts are put together to create a mixed group.For a long time the theory of mixed groups was a neglected area; papers dealing with mixed groups were embarrassingly sparse. The reason was perhaps that no powerful methods were available to deal with more intricate situations. The enormous complexity of their structure made an in-depth study of mixed groups virtually impossible. As a result, the main concern was to find out when a group was not an honest mixed group, but a splitting group: a direct sum of a torsion and a torsion-free group. The characterizations of those torsion and the torsion-free groups that force splitting were the main results on mixed groups up to the 1970s.The theory was suddenly revitalized: the impetus was the revision of the traditional view as an extension of a torsion group by a torsion-free group. Mixed groups could also be regarded in the diagonally opposite way as an extension of a torsion-free group by a torsion group. This point of view, inspired by J. Rotman, turned out a more profound approach to mixed groups, though there was no canonical way of treating them from this angle. New life was brought to the study of mixed groups after it was observed, in addition, that newly developed methods in the theory of torsion groups may be used, mutatis mutandis, to deal with some special interesting mixed groups. The theory of valuated groups—as the new vehicle for research—was developed under the leadership of the New Mexico group theorists.The quintessence of the new approach, first systematically applied by Warfield, is the idea of ignoring temporarily the torsion subgroup, but at the same time enriching a free subgroup of maximal rank with the height functions (as valuations) of its elements. In this way, relevant information about the absent torsion subgroup is incorporated into the free subgroup considered, which sufficed in interesting special cases.In the course of developing a theory of mixed groups, we start with the conventional splitting problem. We then introduce the machinery needed for the discussion of tractable classes of mixed groups. Some tools developed for torsion and torsion-free groups need to be readjusted, and new tools have to be acquired to deal with the new entities. Existence and extension of homomorphisms between mixed groups are in the center of discussion. The highlights are the structure theorems on simply presented and Warfield groups.KeywordsMixed GroupWarfield GroupsTorsion-free RankBaer RadicalComplete Discrete Valuation DomainThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.