Some subgroups of the Burnside group $B_0(2,5)$

A. A. Kuznetsov ; A. S. Kuznetsova

Prikladnaya Diskretnaya Matematika. Supplement, no. 14 (2021), pp. 184-186

Résumé

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) $.

Détail

Keywords: non-commutative cryptography, Burnside group.