On the growth functions of finite two generator burnside groups of exponent five

Let B₀(2, 5) = (a₁,a₂) be the largest 2-generator Burnside group of exponent 5. It has the order 5. There is a power commutator presentation of B₀(2, 5). In this case, every element of the group can be uniquely represented as a^ ' a₂ ' ... ' a₃₄, where a E Z₅, a E B₀(2, 5), i = 1, 2,..., 34. Here, a₁ and a₂ are generators of B₀(2, 5), commutators a₃,..., a₃₄ are recursively defined by a₁ and a₂. We define B_k = B₀(2,5)/(a_k+1,..., a₃₄) as a quotient of B₀(2, 5). It is clearly that |B_k | = 5. A new algorithm for computing the growth function of B_k is created. Using this algorithm, we calculated the growth functions of B_k relative to generating sets {a₁,a₂} and {a₁, a-, a₂, a-} for k = 15, 16, 17.

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