On commutator fully transitive abelian groups. (Q2352892)

The commutator subgroup of an abelian group \(G\) is the additive subgroup of its endomorphism ring \(\text{End}(G)\) generated by all commutators \(\alpha\beta-\beta\alpha\) and its commutator subring is the subring of \(\text{End}(G)\) generated by commutators. The first problem addressed by this paper is the characterization of those groups \(G\) for which the commutator subgroup and subring are equal or are equal to \(\text{End}(G)\). They establish definitive results for divisible groups, bounded \(p\)-groups, completely decomposable groups and vector groups, and several closure properties. The second problem is to characterize those groups \(G\) which are commutator fully transitive or strongly commutator fully transitive in the following sense: for all non-zero \(x\) and \(y\) in \(G\) with \(H(x)\leq H(y)\) and \(o(x)\) divides \(o(y)\) or \(o(x) =\infty\), (where \(H(x)\) is the height matrix or Ulm indicator of \(x\) and \(o(x)\) the order of \(x\)), there exists \(\varphi\) in the commutator subring or commutator subgroup such that \(\varphi(x)=y\). The authors establish that these properties are stronger than the classical concept of full transitivity, but that there are many similarities between these concepts.