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The classical notions of transitivity and full transitivity in Abelian p-groups have natural extensions to concepts called Krylov and weak transitivity. The interconnections between these four types of transitivity are determined for Abelian p-groups; there is a marked difference in the relationships when the prime p is equal to 2. In the final section the relationship between full and Krylov transitivity is examined in the case of mixed Abelian groups which are p-local in the sense that multiplication by an integer relatively prime to p is an automorphism.
Braun, G., Goldsmith, B., Gong, K. & Strungmann, L. (2019). Some transitivity-like concepts in Abelian groups. Journal of Algebra, 529(July), pp.114-123. doi:10.1016/j.jalgebra.2019.03.026