POWERS OF SUBSETS IN FREE PERIODIC GROUPS

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Submitted: 2025-01-22; Published: 2025-01-22

This work is licensed under Creative Commons Attribution–NonCommercial International License (CC BY-NC 4.0).

Abstract

It is proved that for every odd n≥1039 there are two words u(x,y),v(x,y) of length ≤658n2 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 |St|>4⋅2.9[t658s2] holds, where s is the smallest odd divisor of n that satisfies the inequality s≥1039.