On projectively Krylov transitive and projectively weakly transitive abelian \(p\)-groups (Q528909)

Let \(G\) be an abelian \(p\)-group, \(\mathcal E(G)\) its endomorphism ring and \(\mathrm{Proj}(G)\) the subring of \(\mathcal E(G)\) additively generated by its idempotents. The authors call \(G\) projectively Krylov transitive if whenever \(x\) and \(y\) in \(G\) have equal Ulm sequences, there exists \(\phi\in \mathrm{Proj}(G)\) with \(\phi(x)=y\). This condition is then progressively strengthened by requiring \(\phi\) to be merely in the additive group \(\mathrm{Proj}^+(G)\), or an automorphism, or an automorphism in \(\mathrm{Proj}^+(G)\). Thus, the authors define no less than six equivalence relations on \(G\) and then proceed to systematically describe the connections among these relations. As may be expected, there are sharp differences between the cases \(p=2\) and \(p>2\).