  We show that finitely generated Abelian groups admit equal growth functions
with respect to symmetric generating sets if and only if they have the same
rank and the torsion parts have the same parity. In contrast, finitely
generated Abelian groups admit equal growth functions with respect to monoid
generating sets if and only if they have same rank. Moreover, we show that the
size of the torsion part is in fact determined by the set of all growth
functions of a finitely generated Abelian group.
