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This article is cited in 5 scientific papers (total in 5 papers)
Computational Methods in Discrete Mathematics
An algorithm for computation of the growth functions in finite two-generated groups of exponent $5$
A. A. Kuznetsov Siberian State Aerospace University, Krasnoyarsk, Russia
Abstract:
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 $k=15,16,17$.
Keywords:
Burnside group, the growth function.
Citation:
A. A. Kuznetsov, “An algorithm for computation of the growth functions in finite two-generated groups of exponent $5$”, Prikl. Diskr. Mat., 2016, no. 3(33), 116–125
Linking options:
https://www.mathnet.ru/eng/pdm554 https://www.mathnet.ru/eng/pdm/y2016/i3/p116
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