We introduce two new notions of transitivity for Abelian 𝑝-groups based on isomorphism of quotients rather than the classical use of equality of height sequences associated with Abelian 𝑝-group theory.
Unlike the classical theory where “most” groups are transitive, these new notions lead to much smaller classes, but even these classes are sufficiently large to be interesting.
A. L. S. Corner,
The independence of Kaplansky’s notions of transitivity and full transitivity,
Quart. J. Math. Oxford (2) 27 (1976), 15–20.