A resource-efficient algorithm for study the growth in finite two-generator groups of exponent 5 | Prikladnaya Diskretnaya Matematika - Applied Discrete Mathematics. 2018. № 42. DOI: 10.17223/20710410/42/7

For studying the growth in finite groups, we present a resource-efficient algorithm which is a modification of our early algorithm. The purpose of the modification is to minimize the space complexity of the algorithm and to save its time complexity at an acceptable level. The main idea of the modified algorithm is to take in the given group G a suitable subgroup N such that |N| ^ |G|, to calculate growth functions for all cosets gN independently of each other, to summarize these functions and to obtain the growth function for the group G. By using this algorithm, we calculate the growth functions for the group Big with two generators a₁ and a₂ and for the groups Big, B19 with four generators a₁, a-¹, a₂ and a-¹, where B_k = B₀(2, 5)/(a_k+1,..., a₃₄) is a quotient of the group B₀(2, 5) = (a₁,a₂) which is the largest two-generator Burn-side group of exponent 5 (its order is 5³⁴), a₁ and a₂ are generators of B₀ (2, 5) and a₃,..., a₃₄ are the commutators of B₀(2, 5), so each element in B₀(2, 5) can be represented as a^¹ -а^² ·.. .-aOf⁴, a G Z₅, i = 1,2,..., 34. Based on these data, we formulate a hypothesis about the diameters of Cayley graphs of the group B₀(2, 5) with generating sets A₂ = {а₁,а₂} and A₄ = {a₁, a-¹, a₂, a-¹}, namely, Da₂(B₀(2, 5)) w 105 and Da₄(B0(2, 5)) w 69.