  We extend the definition of Jamison sequences in the context of topological
abelian groups. Then we study such sequences when the abelian group is discrete
and countably infinite. An arithmetical characterization of such sequences is
obtained, extending the result of Badea and Grivaux about Jamison sequences of
integers. In particular, we prove that the sequence consisting of all elements
of the group is a Jamison sequence. In the opposite, a sequence which generates
a subgroup of infinite index in the group is never a Jamison sequence.
