  The present lack of a stable method to compare persistent homology groups
with torsion is a relevant problem in current research about Persistent
Homology and its applications in Pattern Recognition. In this paper we
introduce a pseudo-distance d_T that represents a possible solution to this
problem. Indeed, d_T is a pseudo-distance between multidimensional persistent
homology groups with coefficients in an Abelian group, hence possibly having
torsion. Our main theorem proves the stability of the new pseudo-distance with
respect to the change of the filtering function, expressed both with respect to
the max-norm and to the natural pseudo-distance between topological spaces
endowed with vector-valued filtering functions. Furthermore, we prove a result
showing the relationship between d_T and the matching distance in the
1-dimensional case, when the homology coefficients are taken in a field and
hence the comparison can be made.
