This research addresses the problem posed by Chekhlov and Danchev (2015) regarding variations of Kaplansky’s full transitivity in primary abelian groups 𝐺. By delving into three distinct forms of full transitivity within the endomorphism ring of 𝐺, specifically focusing on subgroups, subrings, and unitary subrings generated by commutator endomorphisms, we aim to provide a comprehensive understanding of the totally projective groups exhibiting these properties. The Ulm function of 𝐺 emerges as a key tool in solving this problem and related inquiries, leading to a precise characterization of the groups involved.
Keywords: Kaplansky's notion, commutator endomorphisms., endomorphism ring, full transitivity, primary Abelian groups, totally projective groups, ulm function
