Computational experiments in finite two generator burnside groups of exponent five

Let B₀(2, 5) = (a₁,a₂) be the largest two generator Burnside group of exponent five. It has the order 5³⁴. There is a power commutator presentation of B₀(2, 5). In this case every element of the group can be represented uniquely as a^¹ · a^² · ... · aOf⁴, a_i £ Z₅, i = 1, 2,..., 34. Here a₁ and a₂ are generators of B₀(2, 5), commutators a₃,...,a₃₄ are defined recursively by a₁ and a₂. We define = B₀(2, 5)/(a_k+1,... ,a₃₄) as a quotient of B₀(2, 5), |B_k| = 5^k. Let p be the homomorphism of onto the group and be the kernel of p. We have done some computational experiments and now formulate a hypothesis about the diameter D_A4(B_k) of the relative to the symmetric generating set A₄ = {a₁, a-¹, a₂, a-¹}: D_A4(eN_k) = D_A4(B_k) for all 2 ^ k ^ 34 where |N_k| ~ |Q_k| ~ |B_k|¹/2, e is the identity of and D_A4 (eN_k) is the diameter of the coset eN_k. Note that this hypothesis is correct for k ^ 19.

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