On commutator Krylov transitive and commutator weakly transitive abelian \(p\)-groups (Q2279825)

The authors define the new concepts of commutator (Krylov) transitive and strongly commutator (Krylov) transitive abelian \(p\)-groups which generalize the notions researched by the authors [J. Group Theory 18, No. 4, 623--647 (2015; Zbl 1325.20049)]. The authors prove that: 1) a separable group is commutator Krylov transitive if, and only if, it is commutator fully transitive; 2) if \(G\) is a \(2\)-group of length \(\leq\omega\) for some natural \(n\geq 1\), then \(G\) is commutator Krylov transitive if, and only if, \(G\) is commutator fully transitive; 3) commutator (Krylov, weakly) transitive groups are not closed under taking direct summands and with respect to the formation of (finite or infinite) direct sums. The concept of a commutator weakly transitive abelian \(p\)-group is defined and explored also.