Totally Transitive Torsion-Free Groups of Finite $p$-Rank

Totally transitive torsion-free groups all non-zero endomorphisms of which are monomorphisms are studied. Such are characterized in terms of modules over some rings with specific properties. Also, a structural description is provided for quasihomogeneous, totally transitive, torsion-free groups $A$ of finite $p$-rank, for at least one prime $p$ not dividing $A$.