Abelian groups of zero adjoint entropy

The notion of adjoint entropy for the endomorphisms of an Abelian group is somehow dual to that of algebraic entropy. The Abelian groups of zero adjoint entropy, namely, whose endomorphisms all have zero adjoint entropy, are investigated. Torsion groups and cotorsion groups satisfying this condition are characterized. It is shown that many classes of torsion-free groups contain groups of either zero or innite adjoint entropy. In particular, no characteriza- tion of torsion-free groups of zero adjoint entropy is possible. It is also proved that the mixed groups of a wide class all have innite adjoint entropy.