Classifying some infinite abelian groups and answering Kaplansky's test questions

    Item Description
    Linked Agent
    Creator (cre): Shaw, Alexander F.
    Advisor (adv): Keef, Patrick
    May 8, 2019
    Graduation Year

    In his influential title Infinite Abelian Groups, Irving Kaplansky posed two general questions designed to test classifications of abelian groups. This work answers the questions for a subclass of abelian p-groups that are entirely characterized by their socles (the subgroups with 0 and all elements of order p). The socle is generalized as a valuated vector space and much of this work is dedicated to classifying this generalization in terms of Ulm invariants. For these groups, the questions can thus be translated in two steps: first into the terms of socles and then into the terms of Ulm invariants. The first step is made by Fuchs and Irwin in [1]. This work makes the second step, building up the results and classifications while assuming only working knowledge of introductory algebra. In culmination, this work answers Kaplansky's test questions and gives an example to which the results apply.

    41 pages
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