Geodesic
On the growth functions of finite two generator Burnside groups of exponent five
Prikladnaya Diskretnaya Matematika. Supplement, no. 9 (2016), pp. 132-135.

Voir la notice de l'article provenant de la source Math-Net.Ru

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 presentation 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 $k=15,16,17$.
Keywords: Burnside group, the growth function. 
@article{PDMA_2016_9_a51,
     author = {A. A. Kuznetsov and S. S. Karchevsky},
     title = {On the growth functions of finite two generator {Burnside} groups of exponent five},
     journal = {Prikladnaya Diskretnaya Matematika. Supplement},
     pages = {132--135},
     publisher = {mathdoc},
     number = {9},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDMA_2016_9_a51/}
}
TY  - JOUR
AU  - A. A. Kuznetsov
AU  - S. S. Karchevsky
TI  - On the growth functions of finite two generator Burnside groups of exponent five
JO  - Prikladnaya Diskretnaya Matematika. Supplement
PY  - 2016
SP  - 132
EP  - 135
IS  - 9
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/PDMA_2016_9_a51/
LA  - ru
ID  - PDMA_2016_9_a51
ER  - 
%0 Journal Article
%A A. A. Kuznetsov
%A S. S. Karchevsky
%T On the growth functions of finite two generator Burnside groups of exponent five
%J Prikladnaya Diskretnaya Matematika. Supplement
%D 2016
%P 132-135
%N 9
%I mathdoc
%U http://geodesic.mathdoc.fr/item/PDMA_2016_9_a51/
%G ru
%F PDMA_2016_9_a51
A. A. Kuznetsov; S. S. Karchevsky. On the growth functions of finite two generator Burnside groups of exponent five. Prikladnaya Diskretnaya Matematika. Supplement, no. 9 (2016), pp. 132-135. http://geodesic.mathdoc.fr/item/PDMA_2016_9_a51/

[1] Even S., Goldreich O., “The minimum length generator sequence is NP-hard”, J. Algorithms, 2:3 (1981), 311–313 | DOI | MR | Zbl

[2] Havas G., Wall G., Wamsley J., “The two generator restricted Burnside group of exponent five”, Bull. Austral. Math. Soc., 10 (1974), 459–470 | DOI | MR | Zbl

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

[4] Holt D., Eick B., O'Brien E., Handbook of Computational Group Theory, Chapman Hall/CRC Press, Boca Raton, 2005, 514 pp. | MR | Zbl

[5] Filippov K. A., “O diametre Keli odnoi podgruppy gruppy $B_0(2,5)$”, Vestnik SibGAU, 41:1 (2012), 234–236

[6] Sims C., “Fast multiplication and growth in groups”, Proc. 1998 Intern. Symp. on Symbolic and Algebraic Computation, N.Y., USA, 1998, 165–170 | DOI | Zbl

[7] Kuznetsova A. S., Kuznetsov A. A., Safonov K. V., “Parallelnyi algoritm vychisleniya funktsii rosta v konechnykh dvuporozhdënnykh gruppakh perioda 5”, Prikladnaya diskretnaya matematika. Prilozhenie, 2013, no. 6, 119–121

[8] Kuznetsov A. A., Kuznetsova A. S., “Bystroe umnozhenie elementov v konechnykh dvuporozhdënnykh gruppakh perioda pyat”, Prikladnaya diskretnaya matematika, 2013, no. 1, 110–116