Computational experiments in finite two generator Burnside groups of exponent five

A. A. Kuznetsov

Prikladnaya Diskretnaya Matematika. Supplement, no. 12 (2019), pp. 216-218.

Résumé

Let $B_0(2,5)=\langle a_1,a_2 \rangle$ be the largest two generator Burnside group of exponent five. It has the order $5^{34}$. There is a power commutator presentation of $B_0(2,5)$. In this case every element of the group can be represented uniquely as $a_1^{\alpha_1}\cdot a_2^{\alpha_2}\cdot\ldots\cdot a_{34}^{\alpha_{34}}$, $\alpha_i \in \mathbb{Z}_5$, $i=1,2,\ldots,34$. Here $a_1$ and $a_2$ are generators of $B_0(2,5)$, commutators $a_3,\ldots,a_{34}$ are defined recursively by $a_1$ and $a_2$. We define $B_k=B_0(2,5)/\langle a_{k+1},\ldots,a_{34}\rangle$ as a quotient of $B_0(2,5)$, $|B_k|=5^k$. Let $\varphi $ be the homomorphism of $ B_k $ onto the group $ Q_k $ and $ N_k $ be the kernel of $\varphi $. We have done some computational experiments and now formulate a hypothesis about the diameter $D_{A_4}(B_k)$ of the $B_k$ relative to the symmetric generating set $A_4 = \{ a_1,a_1^{-1},a_2,a_2^{-1}\}$: $D_{A_4}(eN_k) = D_{A_4}(B_k)$ for all $2\leq k \leq 34$ where $|N_k| \sim|Q_k| \sim |B_k|^{{1}/{2}}$, $e$ is the identity of $B_k$ and $D_{A_4}(eN_k)$ is the diameter of the coset $eN_k$. Note that this hypothesis is correct for $k \leq 19$.

Détail

Keywords: Burnside group, the growth function.

Bibliographie

[1] Kuznetsov A. A., “Ob odnom algoritme vychisleniya funktsii rosta v konechnykh dvuporozhdennykh gruppakh perioda 5”, Prikladnaya diskretnaya matematika, 2016, no. 3(33), 116–125

[2] Kuznetsov A. A., Kuznetsova A. S., “Resursno-effektivnyi algoritm dlya issledovaniya rosta v konechnykh dvuporozhdennykh gruppakh perioda 5”, Prikladnaya diskretnaya matematika, 2018, no. 42, 94–103

[3] Konstantinova E. V., Kombinatornye zadachi na grafakh Keli, NGU, Novosibirsk, 2010, 110 pp.

[4] Havas G., Wall G., Wamsley J., “The two generator restricted Burnside group of exponent five”, Bull. Austral. Math. Soc., 1974, no. 10, 459–470 | DOI | MR | Zbl

[5] Sims C., Computation with Finitely Presented Groups, Cambridge University Press, Cambridge, 1994, 628 pp. | MR | Zbl