The main objective of this text is to provide a comprehensive overview of results related to the group minimization problem on a finite Abelian group, also known as the Gomory group minimization problem, and its applications in integer linear programming, combinatorial optimization, and enumerative combinatorics. Significant attention is also given to the problem of computing the generalized convolution of two vectors, as the most advanced and efficient algorithms for solving the group minimization problem heavily rely on convolution computations. From the perspective of applications, particular emphasis is placed on Gomory’s asymptotic algorithm, the integer linear programming problem on the generalized hypercube, and the integer feasibility problem in a simplex.