The group problem of minimization

This problem was shown previously. The algorithm for reduction of an integer basic matrix to a normal form was used in the process. The speedup of calculations in this algorithm is gained due to the Euclidian algorithm. The following questions are considered in this article: (i) the statement of the group problem of minimization for a finite Abelian group; (ii) the numerical solution of this problem; (iii) the representation of the group elements is given for a cyclic group and for a direct sum of cyclic groups; (iiii) recurrence relations for the value function and the index function; (iiiii) calculation of the coefficients of the inequation, giving the facet of the polytope of the group equation; (iiiiii) a statement and a proof of the theorem on the steps number estimation. The recurrence relations and the theorem mentioned are the theoretical basis for the validation of the computational schemes.