CLASSIFICATION OF FULLY TRANSITIVE PRIMARY ABELIAN GROUPS USING ULM INVARIANTS

This study investigates variations of Kaplansky's full transitivity in primary abelian groups, with particular emphasis on their characterisation through Ulm functions. Building on the foundational contributions of Chekhlov and Danchev, the research examines the structural relationships among subgroups, subrings, and unitary subrings generated by commutator endomorphisms within the endomorphism ring of primary abelian groups. The study employs Ulm function theory as a principal analytical framework to investigate the properties of totally projective groups and establish conditions under which these variations of full transitivity hold. Particular attention is given to the role of commutator endomorphisms in determining the algebraic structure and transitivity properties of the groups under consideration. By integrating concepts from abelian group theory, endomorphism rings, and Ulm invariants, the research seeks to provide a comprehensive characterisation of the specified transitivity variations and address the outstanding problems associated with their structural properties. The findings are expected to contribute to a deeper understanding of the relationship between Ulm functions and full transitivity in totally projective primary abelian groups.