Study of the growth functions in two-generator groups of the exponent 5

Let B 0 (2,5, k) be the largest finite two generated Burnside group of the exponent 5 of the nilpotency class k and {α 1, α 2} be generators of this group. Earlier the author together with A.A. Kuznetsov obtained the growth functions of B 0(2,5, k) with respect to the generating set {α 1, α 1- 1α 2, α- 1} for k ≤ 5. In this work a computer algorithm that calculates the growth function and the Cayley diameter of the graph of a finite p-group, defined by the generating set А = {α 1, α 2}, is created. On the basis of the algorithm the growth functions of B 0(2,5, k) with respect to the generating set А for k ≤ 5 is obtained. The considered task, besides for substantial significance, has applications as well. For example, a network of processors for parallel computation can be regarded as an undirected graph in which the vertices are the processors, and two vertices are joined with the edge if there is a direct connection between the two processors represented. The design of a large network is more feasible if the number of connections is small, but computations become more efficient if the short paths connect any two vertices (that is, the diameter of the graph should be as small as possible). Of course, these two requirements tend to conflict with each other. At the group-theoretic language the diameter of the graph of a computing network is equal to the maximum length of minimal words of the group graph.