A resource-efficient algorithm for study the growth in finite two-generator groups of exponent~$5$

A. A. Kuznetsov ; A. S. Kuznetsova

Prikladnaâ diskretnaâ matematika, no. 4 (2018), pp. 94-103

Résumé

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 $|N|\ll|G|$, 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 $B_{18}$ with two generators $a_1$ and $a_2$ and for the groups $B_{18}$, $B_{19}$ with four generators $a_1$, $a_1^{-1}$, $a_2$ and $a_2^{-1}$, where $B_k=B_0(2,5)/\langle a_{k+1},\ldots,a_{34}\rangle$ is a quotient of the group $B_0(2,5)=\langle a_1,a_2 \rangle$ which is the largest two-generator Burnside group of exponent 5 (its order is $5^{34}$), $a_1$ and $a_2$ are generators of $B_0(2, 5)$ and $a_3,\ldots,a_{34}$ are the commutators of $B_0(2, 5)$, so each element in $B_0(2, 5)$ can be represented as $a_1^{\alpha_1}\cdot a_2^{\alpha_2}\cdot\ldots\cdot a_{34}^{\alpha_{34}}$, $\alpha_i \in \mathbb{Z}_5$, $i=1,2,\ldots,34$. Based on these data, we formulate a hypothesis about the diameters of Cayley graphs of the group $B_0(2, 5)$ with generating sets $A_2=\{ a_1,a_2 \}$ and $A_4 = \{ a_1,a_1^{-1},a_2,a_2^{-1}\}$, namely, $D_{A_2}(B_0(2,5)) \approx 105$ and $D_{A_4}(B_0(2,5)) \approx 69$.

Détail

Keywords: Burnside group, the Cayley graph, the growth function.