An algorithm for computation of the growth functions in finite two-generated groups of exponent~$5$

A. A. Kuznetsov

Prikladnaâ diskretnaâ matematika, no. 3 (2016), pp. 116-125

Résumé

Let $B_0(2,5)=\langle a_1,a_2\rangle$ be the largest $2$-generator Burnside group of exponent $5$. It has the order $5^{34}$. There is a power commutator representation of $B_0(2,5)$. In this case, every element of the group can be uniquely represented as $a_1^{\alpha_1}\cdot a_2^{\alpha_2}\cdot\dots\cdot a_{34}^{\alpha_{34}}$, where $\alpha_i \in\mathbb Z_5$, $a_i\in B_0(2,5)$, $i=1,2,\dots,34$. Here, $a_1$ and $a_2$ are generators of $B_0(2,5)$, commutators $a_3,\dots,a_{34}$ are recursively defined by $a_1$ and $a_2$. We define $B_k=B_0(2,5)/\langle a_{k+1},\dots,a_{34}\rangle$ as a quotient of $B_0(2,5)$. It is clearly that $|B_k|=5^k$. 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 $\{a_1,a_2\}$ and $\{a_1,a_1^{-1},a_2,a_2^{-1}\}$ for $k=15,16,17$.

Détail

Keywords: Burnside group, the growth function.