Geodesic
Some subgroups of the Burnside group $B_0(2,5)$
Prikladnaya Diskretnaya Matematika. Supplement, no. 14 (2021), pp. 184-186.

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Let $ B_0(2,5) = \langle x, y \rangle $ be the largest finite two generator Burnside group of exponent five and order $ 5 ^ {34} $. We study a series of subgroups $ H_i = \langle a_i, b_i \rangle $ of the group $ B_0 (2,5) $, where $ a_0 = x $, $ b_0 = y $, $ a_i = a_ { i-1} b_ {i-1} $ and $ b_i = b_ {i-1} a_ {i-1} $ for $ i \in \mathbb {N} $. It has been found that $H_4$ is a commutative group. Therefore, $H_5$ is a cyclyc group and the series of subgroups is broken. The elements $ a_4 = xy ^ 2xyx ^ 2y ^ 2x ^ 2yxy ^ 2x $ and $ b_4 = yx ^ 2yxy ^ 2x ^ 2y ^ 2xyx ^ 2y $ of length $16$ generate an abelian subgroup of order $25$ in $ B_0 (2,5) $. Using computer calculations, we have found that there is no other pair of group words of length less than $16$ that generate a noncyclic abelian subgroup in $ B_0 (2,5) $.
Keywords: non-commutative cryptography, Burnside group. 
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     title = {Some subgroups of the {Burnside} group $B_0(2,5)$},
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A. A. Kuznetsov; A. S. Kuznetsova. Some subgroups of the Burnside group $B_0(2,5)$. Prikladnaya Diskretnaya Matematika. Supplement, no. 14 (2021), pp. 184-186. http://geodesic.mathdoc.fr/item/PDMA_2021_14_a43/

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