POWERS OF SUBSETS IN FREE PERIODIC GROUPS

Varouzhan S. Atabekyan; Hayk T. Aslanyan; Satenik T. Aslanyan

doi:10.46991/PYSU:A/2022.56.2.043

Authors

Varouzhan S. Atabekyan Chair of Algebra and Geometry, YSU, Armenia
Hayk T. Aslanyan American University of Armenia, Armenia
Satenik T. Aslanyan Russian-Armenian University, Armenia

DOI:

https://doi.org/10.46991/PYSU:A/2022.56.2.043

Keywords:

power of subset, product of subset, Burnside group

Abstract

It is proved that for every odd $n \ge 1039$ there are two words $u(x, y), v(x,y)$ of length $\le 658n^2$ over the group alphabet $\{x,y\}$ of the free Burnside group $B(2 ,n),$ which generate a free Burnside subgroup of the group $B(2,n)$. This implies that for any finite subset $S$ of the group $B(m,n)$ the inequality $|S^t|>4\cdot 2.9^{[\frac{t}{658s^2}]}$ holds, where $s$ is the smallest odd divisor of $n$ that satisfies the inequality $s\ge1039$.

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