Computational Analysis of Quantitative Characteristics of Some Residual Properties of Solvable Baumslag–Solitar Groups

Abstract

Let G_k be defined as G_k = $\left\langle {a,b;{{a}^{{ - 1}}}ba = {{b}^{k}}} \right\rangle$, where k ≠ 0. It is known that if p is a prime number then G_k is residually a finite p-group iff p|k – 1. It is also known that if p and q are primes not dividing k – 1, p < q and π = {p, q} then G_k is residually a finite π-group iff (k, q) = 1, p|q – 1 and the order of k in the multiplicative group of the field ℤ_q is a p-number. This paper examines the question of the number of two-element sets of prime numbers that satisfy the conditions of the last criterion. More precisely, let f_k(x) be the number of sets {p, q} such that p < q, p $\nmid$ k – 1, q $\nmid$ k – 1, (k, q) = 1, p|q – 1, the order of k modulo q is a p-number, and p and q are chosen among the first x primes. We state that, if 2 ≤ |k| ≤ 10 000 and 1 ≤ x ≤ 50 000, then for almost all considered k the function f_k(x) can be approximated quite accurately by the function α_kx^0.85, where the coefficient α_k is different for each k and {α_k|2 ≤ |k| ≤ 10 000} ⊆ (0.28; 0.31]. In addition, the dependence of the value f_k(50 000) on k is investigated and an effective algorithm for checking a two-element set of prime numbers for compliance with the conditions of the last criterion is proposed. The results may have applications in the theory of computational complexity and algebraic cryptography.