If an abelian group is given by a finite presentation, there is a well known algorithm that obtains its decomposition into a direct product of cyclic groups by reduction of the relation matrix into Smith normal form. The authors claim to give a more efficient algorithm for the case of a finite abelian p-group but they assume that e.g. orders of certain elements are known or can be computed, i.e. that the elements of the group are known in a form that allows easy computation of products. So the use of the algorithm is restricted essentially to the case that one has to deal with a subgroup of a group for which a base is known. The article contains some undefined notation which must be guessed from the context.