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We define the concepts of  commutator (Krylov) transitive  and  strongly commutator (Krylov) transitive  Abelian  p -groups. These two innovations are respectively non-trivial generalizations of the notions of commutator fully transitive and strongly commutator fully transitive  p -groups from a paper of Chekhlov and Danchev (J. Group Theory, 2015). They are also commutator socle-regular in the sense of Danchev and Goldsmith (J. Group Theory, 2014). Various results from there and from a paper of Goldsmith and Strüngmann (Houston J. Math., 2007) are considerably extended to this new point of view. We also define and explore the concept of a  commutator weakly transitive  Abelian  p -group, comparing its properties with those of the aforementioned two group classes. Some affirmations, sounding quite curiously, are detected in order to illustrate the pathology of the commutators in the endomorphism rings of  p -primary Abelian groups.

On commutator Krylov transitive and commutator weakly transitive Abelian p -groups